Dual space: Difference between revisions
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{{Short description|In mathematics, vector space of linear forms}} | {{Short description|In mathematics, vector space of linear forms}} | ||
{{Use American English|date = March 2019}} | {{Use American English|date = March 2019}} | ||
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== Algebraic dual space == | == Algebraic dual space == | ||
Given any [[vector space]] <math>V</math> over a [[field (mathematics)|field]] <math>F</math>, the '''(algebraic) dual space''' <math>V^{*}</math><ref>{{harvtxt|Katznelson|Katznelson|2008}} p. 37, §2.1.3</ref> (alternatively denoted by <math>V^{\lor}</math><ref name=":03">{{harvtxt|Tu|2011}} p. 19, §3.1</ref> or <math>V'</math><ref>{{harvtxt|Axler|2015}} p. 101, §3.94</ref><ref>{{Harvp|Halmos|1974}} p. 20, §13</ref>)<ref group="nb">For <math>V^{\lor}</math> used in this way, see ''[[ | Given any [[vector space]] <math>V</math> over a [[field (mathematics)|field]] <math>F</math>, the '''(algebraic) dual space''' <math>V^{*}</math><ref>{{harvtxt|Katznelson|Katznelson|2008}} p. 37, §2.1.3</ref> (alternatively denoted by <math>V^{\lor}</math><ref name=":03">{{harvtxt|Tu|2011}} p. 19, §3.1</ref> or <math>V'</math><ref>{{harvtxt|Axler|2015}} p. 101, §3.94</ref><ref>{{Harvp|Halmos|1974}} p. 20, §13</ref>)<ref group="nb">For <math>V^{\lor}</math> used in this way, see ''[[An Introduction to Manifolds]]'' ({{Harvnb|Tu|2011|p=19}}). | ||
This notation is sometimes used when <math>(\cdot)^*</math> is reserved for some other meaning. | This notation is sometimes used when <math>(\cdot)^*</math> is reserved for some other meaning. | ||
For instance, in the above text, <math>F^*</math> is frequently used to denote the codifferential of ''<math>F</math>'', so that <math>F^* \omega</math> represents the pullback of the form <math>\omega</math>. | For instance, in the above text, <math>F^*</math> is frequently used to denote the codifferential of ''<math>F</math>'', so that <math>F^* \omega</math> represents the pullback of the form <math>\omega</math>. | ||
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</ref> is defined as the set of all [[linear map]]s ''<math>\varphi: V \to F</math>'' ([[linear functional]]s). Since linear maps are vector space [[homomorphism]]s, the dual space may be denoted <math>\hom (V, F)</math>.<ref name=":03"/> | </ref> is defined as the set of all [[linear map]]s ''<math>\varphi: V \to F</math>'' ([[linear functional]]s). Since linear maps are vector space [[homomorphism]]s, the dual space may be denoted <math>\hom (V, F)</math>.<ref name=":03"/> | ||
The dual space <math>V^*</math> itself becomes a vector space over ''<math>F</math>'' when equipped with an addition and scalar multiplication satisfying: | The dual space <math>V^*</math> itself becomes a vector space over ''<math>F</math>'' when equipped with an addition and scalar multiplication satisfying: | ||
<math display = "block"> | |||
\begin{align} | \begin{align} | ||
(\varphi + \psi)(x) &= \varphi(x) + \psi(x) \\ | (\varphi + \psi)(x) &= \varphi(x) + \psi(x) \\ | ||
(a \varphi)(x) &= a \left(\varphi(x)\right) | (a \varphi)(x) &= a \left(\varphi(x)\right) | ||
\end{align}</math> | \end{align}</math> | ||
for all <math>\varphi, \psi \in V^*</math>, ''<math>x \in V</math>'', and <math>a \in F</math>. | for all <math>\varphi, \psi \in V^*</math>, ''<math>x \in V</math>'', and <math>a \in F</math>. For example, if we express the vector space <math>\Reals^2</math> as the set of vectors <math>r\mathbf{i} + s\mathbf{j}</math> (where <math>r</math> and <math>s</math> are real numbers), the function | ||
<math display = "block">f(r\mathbf{i} + s\mathbf{j}) = r + s</math> | |||
is an element of <math>(\Reals^2)^*</math>, since it is <math>\Reals</math>-linear and maps vectors in <math>\Reals^2</math> to elements of <math>\Reals</math>. | |||
Elements of the algebraic dual space <math>V^*</math> are sometimes called '''covectors''', '''one-forms''', or '''[[linear form]]s'''. | Elements of the algebraic dual space <math>V^*</math> are sometimes called '''covectors''', '''one-forms''', or '''[[linear form]]s'''. | ||
The pairing of a functional ''<math>\varphi</math>'' in the dual space <math>V^*</math> and an element ''<math>x</math>'' of ''<math>V</math>'' is sometimes denoted by a bracket: ''<math>\varphi (x) = [x, \varphi]</math>''<ref>{{Harvp|Halmos|1974}} p. 21, §14</ref> | The pairing of a functional ''<math>\varphi</math>'' in the dual space <math>V^*</math> and an element ''<math>x</math>'' of ''<math>V</math>'' is sometimes denoted by a bracket: ''<math>\varphi (x) = [x, \varphi]</math>''<ref>{{Harvp|Halmos|1974}} p. 21, §14</ref> | ||
or ''<math>\varphi (x) = \langle x, \varphi \rangle</math>''.<ref>{{harvnb|Misner|Thorne|Wheeler|1973}}</ref> This pairing defines a nondegenerate [[bilinear mapping]]<ref group="nb">In many areas, such as [[quantum mechanics]], | or ''<math>\varphi (x) = \langle x, \varphi \rangle</math>''.<ref>{{harvnb|Misner|Thorne|Wheeler|1973}}</ref> This pairing defines a nondegenerate [[bilinear mapping]]<ref group="nb">In many areas, such as [[quantum mechanics]], <math>\langle \cdot, \cdot \rangle</math> is reserved for a [[sesquilinear form]] defined on <math>V \times V</math>.</ref> <math>\langle \cdot, \cdot \rangle : V \times V^* \to F</math> called the [[natural pairing]]. | ||
===Dual set=== | |||
{{Main article|Dual basis}} | |||
Given a vector space <math>V</math> and a basis <math>E</math> on that space, one can define a [[linearly independent]] set in <math>V^*</math> called the [[dual basis|dual set]]. Each vector in <math>E</math> corresponds to a unique vector in the dual set. This correspondence yields an injection <math>V \to V^*</math>. | |||
If <math>V</math> is finite-dimensional, the dual set is a basis, called the [[dual basis]], and the injection <math>V \to V^*</math> is an [[isomorphism]]. | |||
==== Finite-dimensional case ==== | |||
If <math>V</math> is finite-dimensional, | If <math>V</math> is finite-dimensional and has a basis, <math>\{\mathbf{e}_1,\dots,\mathbf{e}_n\}</math> in <math>V</math>, the dual basis is a set <math>\{\mathbf{e}^1,\dots,\mathbf{e}^n\}</math> of linear functionals on <math>V</math>, defined by the relation | ||
<math display = "block"> \mathbf{e}^i(c^1 \mathbf{e}_1+\cdots+c^n\mathbf{e}_n) = c^i, \quad i=1,\ldots,n </math> | |||
for any choice of coefficients <math>c^i\in F</math>. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations | for any choice of coefficients <math>c^i\in F</math>. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations | ||
<math display = "block"> \mathbf{e}^i(\mathbf{e}_j) = \delta^{i}_{j} </math> | |||
where <math>\delta^{i}_{j}</math> is the [[Kronecker delta]] symbol. This property is referred to as the ''bi-orthogonality property''. | where <math>\delta^{i}_{j}</math> is the [[Kronecker delta]] symbol. This property is referred to as the ''bi-orthogonality property''. | ||
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# The <math>\mathbf{e}^i , i=1, 2, \dots, n, </math> are linear functionals, which map <math> x,y \in V </math> such as <math> x= \alpha_1\mathbf{e}_1 + \dots + \alpha_n\mathbf{e}_n </math> and <math> y = \beta_1\mathbf{e}_1 + \dots + \beta_n \mathbf{e}_n </math> to scalars <math> \mathbf{e}^i(x)=\alpha_i </math> and <math> \mathbf{e}^i(y)=\beta_i</math>. Then also, <math> x+\lambda y=(\alpha_1+\lambda \beta_1)\mathbf{e}_1 + \dots + (\alpha_n+\lambda\beta_n)\mathbf{e}_n </math> and <math> \mathbf{e}^i(x+\lambda y)=\alpha_i+\lambda\beta_i=\mathbf{e}^i(x)+\lambda \mathbf{e}^i(y) </math>. Therefore, <math> \mathbf{e}^i \in V^* </math> for <math> i= 1, 2, \dots, n </math>. | # The <math>\mathbf{e}^i , i=1, 2, \dots, n, </math> are linear functionals, which map <math> x,y \in V </math> such as <math> x= \alpha_1\mathbf{e}_1 + \dots + \alpha_n\mathbf{e}_n </math> and <math> y = \beta_1\mathbf{e}_1 + \dots + \beta_n \mathbf{e}_n </math> to scalars <math> \mathbf{e}^i(x)=\alpha_i </math> and <math> \mathbf{e}^i(y)=\beta_i</math>. Then also, <math> x+\lambda y=(\alpha_1+\lambda \beta_1)\mathbf{e}_1 + \dots + (\alpha_n+\lambda\beta_n)\mathbf{e}_n </math> and <math> \mathbf{e}^i(x+\lambda y)=\alpha_i+\lambda\beta_i=\mathbf{e}^i(x)+\lambda \mathbf{e}^i(y) </math>. Therefore, <math> \mathbf{e}^i \in V^* </math> for <math> i= 1, 2, \dots, n </math>. | ||
# Suppose <math> \lambda_1 \mathbf{e}^1 + \cdots + \lambda_n \mathbf{e}^n =0 \in V^*</math>. Applying this functional on the basis vectors of <math> V </math> successively, lead us to <math> \lambda_1=\lambda_2= \dots=\lambda_n=0 </math> (The functional applied in <math> \mathbf{e}_i </math> results in <math> \lambda_i </math>). Therefore, <math>\{\mathbf{e}^1,\dots,\mathbf{e}^n\}</math> is linearly independent on <math>V^*</math>. | # Suppose <math> \lambda_1 \mathbf{e}^1 + \cdots + \lambda_n \mathbf{e}^n =0 \in V^*</math>. Applying this functional on the basis vectors of <math> V </math> successively, lead us to <math> \lambda_1=\lambda_2= \dots=\lambda_n=0 </math> (The functional applied in <math> \mathbf{e}_i </math> results in <math> \lambda_i </math>). Therefore, <math>\{\mathbf{e}^1,\dots,\mathbf{e}^n\}</math> is linearly independent on <math>V^*</math>. | ||
#Lastly, consider <math> g \in V^* </math>. Then | #Lastly, consider <math> g \in V^* </math>. Then <math display = "block"> | ||
g(x)=g(\alpha_1\mathbf{e}_1 + \dots + \alpha_n\mathbf{e}_n)=\alpha_1g(\mathbf{e}_1) + \dots + \alpha_ng(\mathbf{e}_n)=\mathbf{e}^1(x)g(\mathbf{e}_1) + \dots + \mathbf{e}^n(x)g(\mathbf{e}_n) | g(x)=g(\alpha_1\mathbf{e}_1 + \dots + \alpha_n\mathbf{e}_n)=\alpha_1g(\mathbf{e}_1) + \dots + \alpha_ng(\mathbf{e}_n)=\mathbf{e}^1(x)g(\mathbf{e}_1) + \dots + \mathbf{e}^n(x)g(\mathbf{e}_n) | ||
</math> | </math> so <math>g=g(\mathbf{e}_1)\mathbf{e}^1 + \dots + g(\mathbf{e}_n)\mathbf{e}^n | ||
</math>. So <math>\{\mathbf{e}^1,\dots,\mathbf{e}^n\}</math> generates <math>V^*</math>. | |||
Hence, it is a basis of <math> V^*</math>. | |||
{{Collapse bottom}} | {{Collapse bottom}} | ||
For example, if <math>V</math> is <math>\R^2</math>, let its basis be chosen as <math>\{\mathbf{e}_1=(1/2,1/2),\mathbf{e}_2=(0,1)\}</math>. The basis vectors are not orthogonal to each other. Then, <math>\mathbf{e}^1</math> and <math>\mathbf{e}^2</math> are [[one-form]]s (functions that map a vector to a scalar) such that <math>\mathbf{e}^1(\mathbf{e}_1)=1</math>, <math>\mathbf{e}^1(\mathbf{e}_2)=0</math>, <math>\mathbf{e}^2(\mathbf{e}_1)=0</math>, and <math>\mathbf{e}^2(\mathbf{e}_2)=1</math>. (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as | For example, if <math>V</math> is <math>\R^2</math>, let its basis be chosen as <math>\{\mathbf{e}_1=(1/2,1/2),\mathbf{e}_2=(0,1)\}</math>. The basis vectors are not orthogonal to each other. Then, <math>\mathbf{e}^1</math> and <math>\mathbf{e}^2</math> are [[one-form]]s (functions that map a vector to a scalar) such that <math>\mathbf{e}^1(\mathbf{e}_1)=1</math>, <math>\mathbf{e}^1(\mathbf{e}_2)=0</math>, <math>\mathbf{e}^2(\mathbf{e}_1)=0</math>, and <math>\mathbf{e}^2(\mathbf{e}_2)=1</math>. (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as | ||
<math display = "block"> | |||
\begin{bmatrix} | \begin{bmatrix} | ||
e^{11} & e^{12} \\ | e^{11} & e^{12} \\ | ||
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In general, when <math>V</math> is <math>\R^n</math>, if <math>E=[\mathbf{e}_1|\cdots|\mathbf{e}_n]</math> is a matrix whose columns are the basis vectors and <math>\hat{E}=[\mathbf{e}^1|\cdots|\mathbf{e}^n]</math> is a matrix whose columns are the dual basis vectors, then | In general, when <math>V</math> is <math>\R^n</math>, if <math>E=[\mathbf{e}_1|\cdots|\mathbf{e}_n]</math> is a matrix whose columns are the basis vectors and <math>\hat{E}=[\mathbf{e}^1|\cdots|\mathbf{e}^n]</math> is a matrix whose columns are the dual basis vectors, then | ||
<math display = "block">\hat{E}^\textrm{T}\cdot E = I_n,</math> | |||
where <math>I_n</math> is the [[identity matrix]] of order <math>n</math>. The biorthogonality property of these two basis sets allows any point <math>\mathbf{x}\in V</math> to be represented as | where <math>I_n</math> is the [[identity matrix]] of order <math>n</math>. The biorthogonality property of these two basis sets allows any point <math>\mathbf{x}\in V</math> to be represented as | ||
:<math>\mathbf{x} = \sum_i \langle\mathbf{x},\mathbf{e}^i \rangle \mathbf{e}_i = \sum_i \langle \mathbf{x}, \mathbf{e}_i \rangle \mathbf{e}^i,</math> | : <math>\mathbf{x} = \sum_i \langle\mathbf{x},\mathbf{e}^i \rangle \mathbf{e}_i = \sum_i \langle \mathbf{x}, \mathbf{e}_i \rangle \mathbf{e}^i,</math> | ||
even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product <math>\langle \cdot, \cdot \rangle</math> and the corresponding duality pairing are introduced, as described below in ''{{section link||Bilinear_products_and_dual_spaces}}''. | even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product <math>\langle \cdot, \cdot \rangle</math> and the corresponding duality pairing are introduced, as described below in ''{{section link||Bilinear_products_and_dual_spaces}}''. | ||
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More generally, if <math>V</math> is a vector space of any dimension, then the [[level sets]] of a linear functional in <math>V^*</math> are parallel hyperplanes in <math>V</math>, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.<ref>{{harvnb|Misner|Thorne|Wheeler|1973|loc=§2.5}}</ref> | More generally, if <math>V</math> is a vector space of any dimension, then the [[level sets]] of a linear functional in <math>V^*</math> are parallel hyperplanes in <math>V</math>, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.<ref>{{harvnb|Misner|Thorne|Wheeler|1973|loc=§2.5}}</ref> | ||
=== Infinite-dimensional case === | ==== Infinite-dimensional case ==== | ||
If <math>V</math> is not finite-dimensional but has a [[basis (linear algebra)|basis]]<ref group=nb name="choice">Several assertions in this article require the [[axiom of choice]] for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that <math>\R^\N</math> has a basis. | If <math>V</math> is not finite-dimensional but has a [[basis (linear algebra)|basis]]<ref group=nb name="choice">Several assertions in this article require the [[axiom of choice]] for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that <math>\R^\N</math> has a basis. | ||
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The dual space of <math>\R^\infty</math> is (isomorphic to) <math>\R^\N</math>, the space of ''all'' sequences of real numbers: each real sequence <math>(a_n)</math> defines a function where the element <math>(x_n)</math> of <math>\R^\infty</math> is sent to the number | The dual space of <math>\R^\infty</math> is (isomorphic to) <math>\R^\N</math>, the space of ''all'' sequences of real numbers: each real sequence <math>(a_n)</math> defines a function where the element <math>(x_n)</math> of <math>\R^\infty</math> is sent to the number | ||
<math display = "block">\sum_n a_nx_n,</math> | |||
which is a finite sum because there are only finitely many nonzero <math>x_n</math>. The [[dimension (vector space)|dimension]] of <math>\R^\infty</math> is [[countably infinite]], whereas <math>\R^\N</math> does not have a countable basis. | which is a finite sum because there are only finitely many nonzero <math>x_n</math>. The [[dimension (vector space)|dimension]] of <math>\R^\infty</math> is [[countably infinite]], whereas <math>\R^\N</math> does not have a countable basis. | ||
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This observation generalizes to any<ref group=nb name="choice"/> infinite-dimensional vector space <math>V</math> over any field <math>F</math>: a choice of basis <math>\{\mathbf{e}_\alpha:\alpha\in A\}</math> identifies <math>V</math> with the space <math>(F^A)_0</math> of functions <math>f:A\to F</math> such that <math>f_\alpha=f(\alpha)</math> is nonzero for only finitely many <math>\alpha\in A</math>, where such a function <math>f</math> is identified with the vector | This observation generalizes to any<ref group=nb name="choice"/> infinite-dimensional vector space <math>V</math> over any field <math>F</math>: a choice of basis <math>\{\mathbf{e}_\alpha:\alpha\in A\}</math> identifies <math>V</math> with the space <math>(F^A)_0</math> of functions <math>f:A\to F</math> such that <math>f_\alpha=f(\alpha)</math> is nonzero for only finitely many <math>\alpha\in A</math>, where such a function <math>f</math> is identified with the vector | ||
<math display = "block">\sum_{\alpha\in A} f_\alpha\mathbf{e}_\alpha</math> | |||
in <math>V</math> (the sum is finite by the assumption on <math>f</math>, and any <math>v\in V</math> may be written uniquely in this way by the definition of the basis). | in <math>V</math> (the sum is finite by the assumption on <math>f</math>, and any <math>v\in V</math> may be written uniquely in this way by the definition of the basis). | ||
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The dual space of <math>V</math> may then be identified with the space <math>F^A</math> of ''all'' functions from <math>A</math> to <math>F</math>: a linear functional <math>T</math> on <math>V</math> is uniquely determined by the values <math>\theta_\alpha=T(\mathbf{e}_\alpha)</math> it takes on the basis of <math>V</math>, and any function <math>\theta:A\to F</math> (with <math>\theta(\alpha)=\theta_\alpha</math>) defines a linear functional <math>T</math> on <math>V</math> by | The dual space of <math>V</math> may then be identified with the space <math>F^A</math> of ''all'' functions from <math>A</math> to <math>F</math>: a linear functional <math>T</math> on <math>V</math> is uniquely determined by the values <math>\theta_\alpha=T(\mathbf{e}_\alpha)</math> it takes on the basis of <math>V</math>, and any function <math>\theta:A\to F</math> (with <math>\theta(\alpha)=\theta_\alpha</math>) defines a linear functional <math>T</math> on <math>V</math> by | ||
<math display = "block">T\left (\sum_{\alpha\in A} f_\alpha \mathbf{e}_\alpha\right) = \sum_{\alpha \in A} f_\alpha T(e_\alpha) = \sum_{\alpha\in A} f_\alpha \theta_\alpha.</math> | |||
Again, the sum is finite because <math>f_\alpha</math> is nonzero for only finitely many <math>\alpha</math>. | Again, the sum is finite because <math>f_\alpha</math> is nonzero for only finitely many <math>\alpha</math>. | ||
The set <math>(F^A)_0</math> may be identified (essentially by definition) with the [[Direct sum of modules|direct sum]] of infinitely many copies of <math>F</math> (viewed as a 1-dimensional vector space over itself) indexed by <math>A</math>, i.e. there are linear isomorphisms | The set <math>(F^A)_0</math> may be identified (essentially by definition) with the [[Direct sum of modules|direct sum]] of infinitely many copies of <math>F</math> (viewed as a 1-dimensional vector space over itself) indexed by <math>A</math>, i.e. there are linear isomorphisms | ||
<math display = "block"> V\cong (F^A)_0\cong\bigoplus_{\alpha\in A} F.</math> | |||
On the other hand, <math>F^A</math> is (again by definition), the [[direct product]] of infinitely many copies of <math>F</math> indexed by <math>A</math>, and so the identification | On the other hand, <math>F^A</math> is (again by definition), the [[direct product]] of infinitely many copies of <math>F</math> indexed by <math>A</math>, and so the identification | ||
<math display = "block">V^* \cong \left (\bigoplus_{\alpha\in A}F\right )^* \cong \prod_{\alpha\in A}F^* \cong \prod_{\alpha\in A}F \cong F^A</math> | |||
is a special case of a [[Direct sum of modules#Properties|general result]] relating direct sums (of [[module (mathematics)|module]]s) to direct products. | is a special case of a [[Direct sum of modules#Properties|general result]] relating direct sums (of [[module (mathematics)|module]]s) to direct products. | ||
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If <math>V</math> is an infinite-dimensional <math>F</math>-vector space, the arithmetical properties of [[cardinal numbers]] implies that | If <math>V</math> is an infinite-dimensional <math>F</math>-vector space, the arithmetical properties of [[cardinal numbers]] implies that | ||
<math display = "block">\mathrm{dim}(V)=|A|<|F|^{|A|}=|V^\ast|=\mathrm{max}(|\mathrm{dim}(V^\ast)|, |F|),</math> | |||
where cardinalities are denoted as [[absolute value]]s. For proving that <math>\mathrm{dim}(V)< \mathrm{dim}(V^*),</math> it suffices to prove that <math>|F|\le |\mathrm{dim}(V^\ast)|,</math> which can be done with an argument similar to [[Cantor's diagonal argument]].<ref>{{cite book|title=Elements of mathematics: Algebra I, Chapters 1 - 3|author=Nicolas Bourbaki|page=400|editor=Hermann|isbn=0201006391|year=1974|publisher=Addison-Wesley Publishing Company |language=en}}</ref> The exact dimension of the dual is given by the [[Erdős–Kaplansky theorem]]. | where cardinalities are denoted as [[absolute value]]s. For proving that <math>\mathrm{dim}(V)< \mathrm{dim}(V^*),</math> it suffices to prove that <math>|F|\le |\mathrm{dim}(V^\ast)|,</math> which can be done with an argument similar to [[Cantor's diagonal argument]].<ref>{{cite book|title=Elements of mathematics: Algebra I, Chapters 1 - 3|author=Nicolas Bourbaki|page=400|editor=Hermann|isbn=0201006391|year=1974|publisher=Addison-Wesley Publishing Company |language=en}}</ref> The exact dimension of the dual is given by the [[Erdős–Kaplansky theorem]]. | ||
=== Bilinear products and dual spaces === | === Bilinear products and dual spaces === | ||
If | If <math>V</math> is finite-dimensional, then <math>V</math> is isomorphic to <math>V^*</math>. But there is in general no [[natural isomorphism]] between these two spaces.<ref>{{harvnb|Mac Lane|Birkhoff|1999|loc=§VI.4}}</ref> Any [[bilinear form]] <math>\langle \cdot, \cdot \rangle</math> on <math>V</math> gives a mapping of <math>V</math> into its dual space via | ||
<math display = "block">v\mapsto \langle v, \cdot\rangle</math> | |||
where the right hand side is defined as the functional on | where the right hand side is defined as the functional on <math>V</math> taking each <math>w \in V</math> to <math>\langle v, w \rangle</math>. In other words, the bilinear form determines a linear mapping | ||
<math display = "block">\Phi_{\langle\cdot,\cdot\rangle} : V\to V^*</math> | |||
defined by | defined by | ||
<math display = "block">\left[\Phi_{\langle\cdot,\cdot\rangle}(v), w\right] = \langle v, w\rangle.</math> | |||
If the bilinear form is [[nondegenerate form|nondegenerate]], then this is an isomorphism onto a subspace of | If the bilinear form is [[nondegenerate form|nondegenerate]], then this is an isomorphism onto a subspace of <math>V^*</math>. | ||
If | If <math>V</math> is finite-dimensional, then this is an isomorphism onto all of <math>V^*</math>. Conversely, any isomorphism <math>\Phi</math> from <math>V</math> to a subspace of <math>V^*</math> (resp., all of <math>V^*</math> if <math>V</math> is finite dimensional) defines a unique nondegenerate bilinear form <math> \langle \cdot, \cdot \rangle_{\Phi} </math> on <math>V</math> by | ||
<math display = "block"> \langle v, w \rangle_\Phi = (\Phi (v))(w) = [\Phi (v), w].\,</math> | |||
Thus there is a one-to-one correspondence between isomorphisms of | Thus there is a one-to-one correspondence between isomorphisms of <math>V</math> to a subspace of (resp., all of) <math>V^*</math> and nondegenerate bilinear forms on <math>V</math>. | ||
If the vector space | If the vector space <math>V</math> is over the [[complex numbers|complex]] field, then sometimes it is more natural to consider [[sesquilinear form]]s instead of bilinear forms. | ||
In that case, a given sesquilinear form | In that case, a given sesquilinear form <math>\langle \cdot, \cdot \rangle</math> determines an isomorphism of <math>V</math> with the [[Complex conjugate vector space|complex conjugate]] of the dual space | ||
<math display = "block"> | |||
\Phi_{\langle \cdot, \cdot \rangle} : V\to \overline{V^*}. | \Phi_{\langle \cdot, \cdot \rangle} : V\to \overline{V^*}. | ||
</math> | </math> | ||
The conjugate of the dual space <math>\overline{V^*}</math> can be identified with the set of all additive complex-valued functionals | The conjugate of the dual space <math>\overline{V^*}</math> can be identified with the set of all additive complex-valued functionals <math>f: V \to \Complex</math> such that | ||
<math display = "block"> | |||
f(\alpha v) = \overline{\alpha}f(v). | f(\alpha v) = \overline{\alpha}f(v). | ||
</math> | </math> | ||
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<!-- [[matrix (mathematics)]] and [[transpose]] link here --> | <!-- [[matrix (mathematics)]] and [[transpose]] link here --> | ||
{{Main|Transpose of a linear map}} | {{Main|Transpose of a linear map}} | ||
If | If <math>f: V \to W</math> is a [[linear map]], then the ''[[Transpose#Transpose of a linear map|transpose]]'' (or ''dual'') <math>f^*: W^* \to V^*</math> is defined by | ||
<math display = "block"> | |||
f^*(\varphi) = \varphi \circ f \, | f^*(\varphi) = \varphi \circ f \, | ||
</math> | </math> | ||
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The following identity holds for all ''<math>\varphi \in W^*</math>'' and ''<math>v \in V</math>'': | The following identity holds for all ''<math>\varphi \in W^*</math>'' and ''<math>v \in V</math>'': | ||
<math display = "block"> | |||
[f^*(\varphi),\, v] = [\varphi,\, f(v)], | [f^*(\varphi),\, v] = [\varphi,\, f(v)], | ||
</math> | </math> | ||
where the bracket [ | where the bracket <math>[\cdot,\cdot]</math> on the left is the natural pairing of <math>V</math> with its dual space, and that on the right is the natural pairing of <math>W</math> with its dual. This identity characterizes the transpose,<ref>{{Harvp|Halmos|1974}} §44</ref> and is formally similar to the definition of the [[adjoint of an operator|adjoint]]. | ||
The assignment | The assignment <math>f \mapsto f^*</math> produces an [[injective]] linear map between the space of linear operators from <math>V</math> to <math>W</math> and the space of linear operators from <math>W^*</math> to <math>V^*</math>; this homomorphism is an [[isomorphism]] if and only if <math>W</math> is finite-dimensional. | ||
If | If <math>V = W</math> then the space of linear maps is actually an [[algebra over a field|algebra]] under [[composition of maps]], and the assignment is then an [[antihomomorphism]] of algebras, meaning that <math>{(fg)}^* = g^*f^*</math>. | ||
In the language of [[category theory]], taking the dual of vector spaces and the transpose of linear maps is therefore a [[contravariant functor]] from the category of vector spaces over | In the language of [[category theory]], taking the dual of vector spaces and the transpose of linear maps is therefore a [[contravariant functor]] from the category of vector spaces over <math>F</math> to itself. | ||
It is possible to identify | It is possible to identify <math>f^{**}</math> with <math>f</math> using the natural injection into the double dual. | ||
If the linear map | If the linear map <math>f</math> is represented by the [[matrix (mathematics)|matrix]] <math>A</math> with respect to two bases of <math>V</math> and <math>W</math>, then <math>f^*</math> is represented by the [[transpose]] matrix <math>A^T</math> with respect to the dual bases of <math>W^*</math> and <math>V^*</math>, hence the name. | ||
Alternatively, as | Alternatively, as <math>f</math> is represented by <math>A</math> acting on the left on column vectors, <math>f^*</math> is represented by the same matrix acting on the right on row vectors. | ||
These points of view are related by the canonical inner product on | These points of view are related by the canonical inner product on <math>\Reals^n</math>, which identifies the space of column vectors with the dual space of row vectors. | ||
=== Quotient spaces and annihilators === | === Quotient spaces and annihilators === | ||
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The annihilator of the zero vector is the whole dual space: <math>\{ 0 \}^0 = V^*</math>, and the annihilator of the whole space is just the zero covector: <math>V^0 = \{ 0 \} \subseteq V^*</math>. | The annihilator of the zero vector is the whole dual space: <math>\{ 0 \}^0 = V^*</math>, and the annihilator of the whole space is just the zero covector: <math>V^0 = \{ 0 \} \subseteq V^*</math>. | ||
Furthermore, the assignment of an annihilator to a subset of <math>V</math> reverses inclusions, so that if <math>\{ 0 \} \subseteq S\subseteq T\subseteq V</math>, then | Furthermore, the assignment of an annihilator to a subset of <math>V</math> reverses inclusions, so that if <math>\{ 0 \} \subseteq S\subseteq T\subseteq V</math>, then | ||
<math display = "block"> | |||
\{ 0 \} \subseteq T^0 \subseteq S^0 \subseteq V^* . | \{ 0 \} \subseteq T^0 \subseteq S^0 \subseteq V^* . | ||
</math> | </math> | ||
If <math>A</math> and <math>B</math> are two subsets of <math>V</math> then | If <math>A</math> and <math>B</math> are two subsets of <math>V</math> then | ||
<math display = "block"> | |||
A^0 + B^0 \subseteq (A \cap B)^0 . | A^0 + B^0 \subseteq (A \cap B)^0 . | ||
</math> | </math> | ||
If <math>(A_i)_{i\in I}</math> is any family of subsets of <math>V</math> indexed by <math>i</math> belonging to some index set <math>I</math>, then | If <math>(A_i)_{i\in I}</math> is any family of subsets of <math>V</math> indexed by <math>i</math> belonging to some index set <math>I</math>, then | ||
<math display = "block"> | |||
\left( \bigcup_{i\in I} A_i \right)^0 = \bigcap_{i\in I} A_i^0 . | \left( \bigcup_{i\in I} A_i \right)^0 = \bigcap_{i\in I} A_i^0 . | ||
</math> | </math> | ||
In particular if <math>A</math> and <math>B</math> are subspaces of <math>V</math> then | In particular if <math>A</math> and <math>B</math> are subspaces of <math>V</math> then | ||
<math display = "block"> | |||
(A + B)^0 = A^0 \cap B^0 | (A + B)^0 = A^0 \cap B^0 | ||
</math> | </math> | ||
and<ref group=nb name="choice"/> | and<ref group=nb name="choice"/> | ||
<math display = "block"> | |||
(A \cap B)^0 = A^0 + B^0 . | (A \cap B)^0 = A^0 + B^0 . | ||
</math> | </math> | ||
If <math>V</math> is finite-dimensional and <math>W</math> is a [[vector subspace]], then | If <math>V</math> is finite-dimensional and <math>W</math> is a [[vector subspace]], then | ||
<math display = "block"> | |||
W^{00} = W | W^{00} = W | ||
</math> | </math> | ||
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=== Dimensional analysis === | === Dimensional analysis === | ||
The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector <math>v \in V</math> can be paired with a covector <math>\varphi \in V^*</math> by the natural pairing | The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector <math>v \in V</math> can be paired with a covector <math>\varphi \in V^*</math> by the natural pairing | ||
<math>\langle x, \varphi \rangle := \varphi (x) \in F</math> to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to [[Fraction#Reduction|reducing a fraction]]. Thus while the direct sum <math>V \oplus V^*</math> is a | <math>\langle x, \varphi \rangle := \varphi (x) \in F</math> to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to [[Fraction#Reduction|reducing a fraction]]. Thus while the direct sum <math>V \oplus V^*</math> is a <math>2n</math>-dimensional space (if <math>V</math> is <math>n</math>-dimensional), <math>V^*</math> behaves as an <math>-n</math>-dimensional space, in the sense that its dimensions can be canceled against the dimensions of <math>V</math>. This is formalized by [[tensor contraction]]. | ||
This arises in physics via [[dimensional analysis]], where the dual space has inverse units.<ref>{{cite web | This arises in physics via [[dimensional analysis]], where the dual space has inverse units.<ref>{{cite web | ||
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|date=2012-12-29 | |date=2012-12-29 | ||
|quote=Similarly, one can define <math>V^{T^{-1}}</math> as the dual space to <math>V^T</math> ... | |quote=Similarly, one can define <math>V^{T^{-1}}</math> as the dual space to <math>V^T</math> ... | ||
}}</ref> Under the natural pairing, these units cancel, and the resulting scalar value is [[dimensionless]], as expected. For example, in (continuous) [[Fourier analysis]], or more broadly [[time–frequency analysis]]:<ref group="nb">To be precise, continuous Fourier analysis studies the space of [[Functional (mathematics)|functionals]] with domain a vector space and the space of functionals on the dual vector space.</ref> given a one-dimensional vector space with a [[unit of time]] | }}</ref> Under the natural pairing, these units cancel, and the resulting scalar value is [[dimensionless]], as expected. For example, in (continuous) [[Fourier analysis]], or more broadly [[time–frequency analysis]]:<ref group="nb">To be precise, continuous Fourier analysis studies the space of [[Functional (mathematics)|functionals]] with domain a vector space and the space of functionals on the dual vector space.</ref> given a one-dimensional vector space with a [[unit of time]] <math>t</math>, the dual space has units of [[frequency]]: occurrences ''per'' unit of time (units of <math>1/t</math>). For example, if time is measured in [[second]]s, the corresponding dual unit is the [[inverse second]]: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to <math>3s \cdot 2s^{-1} = 6</math>. Similarly, if the primal space measures length, the dual space measures [[inverse length]]. | ||
== Continuous dual space ==<!-- This section is linked from [[Reflexive space]] --> | == Continuous dual space ==<!-- This section is linked from [[Reflexive space]] --> | ||
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=== Properties === | === Properties === | ||
If | If <math>X</math> is a [[Hausdorff space|Hausdorff]] [[topological vector space]] (TVS), then the continuous dual space of <math>X</math> is identical to the continuous dual space of the [[Complete topological vector space|completion]] of <math>X</math>.{{sfn | Narici|Beckenstein | 2011 | pp=225-273}} | ||
=== Topologies on the dual === | === Topologies on the dual === | ||
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This gives the topology on <math>V</math> of [[uniform convergence]] on sets from <math>\mathcal{A},</math> or what is the same thing, the topology generated by [[seminorm]]s of the form | This gives the topology on <math>V</math> of [[uniform convergence]] on sets from <math>\mathcal{A},</math> or what is the same thing, the topology generated by [[seminorm]]s of the form | ||
<math display = "block">\|\varphi\|_A = \sup_{x\in A} |\varphi(x)|,</math> | |||
where <math>\varphi</math> is a continuous linear functional on <math>V</math>, and <math>A</math> runs over the class <math>\mathcal{A}.</math> | where <math>\varphi</math> is a continuous linear functional on <math>V</math>, and <math>A</math> runs over the class <math>\mathcal{A}.</math> | ||
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This means that a net of functionals <math>\varphi_i</math> tends to a functional <math>\varphi</math> in <math>V'</math> if and only if | This means that a net of functionals <math>\varphi_i</math> tends to a functional <math>\varphi</math> in <math>V'</math> if and only if | ||
<math display = "block">\text{ for all } A\in\mathcal{A}\qquad \|\varphi_i-\varphi\|_A = \sup_{x\in A} |\varphi_i(x)-\varphi(x)|\underset{i\to\infty}{\longrightarrow} 0. </math> | |||
Usually (but not necessarily) the class <math>\mathcal{A}</math> is supposed to satisfy the following conditions: | Usually (but not necessarily) the class <math>\mathcal{A}</math> is supposed to satisfy the following conditions: | ||
* Each point <math>x</math> of <math>V</math> belongs to some set <math>A\in\mathcal{A}</math> | * Each point <math>x</math> of <math>V</math> belongs to some set <math>A\in\mathcal{A}</math>. | ||
* Each two sets <math>A \in \mathcal{A}</math> and <math>B \in \mathcal{A}</math> are contained in some set <math>C \in \mathcal{A}</math>. | |||
* Each two sets <math>A \in \mathcal{A}</math> and <math>B \in \mathcal{A}</math> are contained in some set <math>C \in \mathcal{A}</math> | * <math>\mathcal{A}</math> is closed under the operation of multiplication by scalars. | ||
* <math>\mathcal{A}</math> is closed under the operation of multiplication by scalars | |||
If these requirements are fulfilled then the corresponding topology on <math>V'</math> is Hausdorff and the sets | If these requirements are fulfilled then the corresponding topology on <math>V'</math> is Hausdorff and the sets | ||
<math display = "block">U_A ~=~ \left \{ \varphi \in V' ~:~ \quad \|\varphi\|_A < 1 \right \},\qquad \text{ for } A \in \mathcal{A}</math> | |||
form its local base. | form its local base. | ||
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* The [[Strong topology (polar topology)|strong topology]] on <math>V'</math> is the topology of uniform convergence on [[Bounded set (topological vector space)|bounded subsets]] in <math>V</math> (so here <math>\mathcal{A}</math> can be chosen as the class of all bounded subsets in <math>V</math>). | * The [[Strong topology (polar topology)|strong topology]] on <math>V'</math> is the topology of uniform convergence on [[Bounded set (topological vector space)|bounded subsets]] in <math>V</math> (so here <math>\mathcal{A}</math> can be chosen as the class of all bounded subsets in <math>V</math>). | ||
If <math>V</math> is a [[normed vector space]] (for example, a [[Banach space]] or a [[Hilbert space]]) then the strong topology on <math>V'</math> is normed (in fact a Banach space if the field of scalars is complete), with the norm | If <math>V</math> is a [[normed vector space]] (for example, a [[Banach space]] or a [[Hilbert space]]) then the strong topology on <math>V'</math> is normed (in fact a Banach space if the field of scalars is complete), with the norm | ||
<math display = "block">\|\varphi\| = \sup_{\|x\| \le 1 } |\varphi(x)|.</math> | |||
* The [[stereotype space|stereotype topology]] on <math>V'</math> is the topology of uniform convergence on [[Totally bounded space|totally bounded sets]] in <math>V</math> (so here <math>\mathcal{A}</math> can be chosen as the class of all totally bounded subsets in <math>V</math>). | * The [[stereotype space|stereotype topology]] on <math>V'</math> is the topology of uniform convergence on [[Totally bounded space|totally bounded sets]] in <math>V</math> (so here <math>\mathcal{A}</math> can be chosen as the class of all totally bounded subsets in <math>V</math>). | ||
* The [[weak topology]] on <math>V'</math> is the topology of uniform convergence on finite subsets in <math>V</math> (so here <math>\mathcal{A}</math> can be chosen as the class of all finite subsets in <math>V</math>). | * The [[weak topology]] on <math>V'</math> is the topology of uniform convergence on finite subsets in <math>V</math> (so here <math>\mathcal{A}</math> can be chosen as the class of all finite subsets in <math>V</math>). | ||
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=== Examples === | === Examples === | ||
Let 1 < | Let 1 < <math>p</math> < ∞ be a real number and consider the Banach space ''[[Lp space#The p-norm in countably infinite dimensions|ℓ<sup> p</sup>]]'' of all [[sequence]]s <math>\mathbb{a} = a_n</math> for which | ||
<math display = "block">\|\mathbf{a}\|_p = \left ( \sum_{n=0}^\infty |a_n|^p \right) ^{\frac{1}{p}} < \infty.</math> | |||
Define the number | Define the number <math>q</math> by <math>1/p + 1/q = 1</math>. Then the continuous dual of <math>\ell^p</math> is naturally identified with <math>\ell^q</math>: given an element <math>\varphi \in (\ell^p)'</math>, the corresponding element of <math>\ell^q</math> is the sequence <math>(\varphi(\mathbf {e}_n))</math> where '''<math>\mathbf {e}_n</math>''' denotes the sequence whose <math>n</math>-th term is 1 and all others are zero. Conversely, given an element <math>\mathbb{a} = (a_n) \in \ell^q</math>, the corresponding continuous linear functional ''<math>\varphi</math>'' on <math>\ell^p</math> is defined by | ||
<math display = "block">\varphi (\mathbf{b}) = \sum_n a_n b_n</math> | |||
for all { | for all <math>\mathbb{b} = (b_n) \in \ell^p</math> (see [[Hölder's inequality]]). | ||
In a similar manner, the continuous dual of | In a similar manner, the continuous dual of <math>\ell^1</math> is naturally identified with <math>\ell^\infty</math> (the space of bounded sequences). | ||
Furthermore, the continuous duals of the Banach spaces | Furthermore, the continuous duals of the Banach spaces <math>c</math> (consisting of all [[limit of a sequence|convergent]] sequences, with the [[supremum norm]]) and <math>c_0</math> (the sequences converging to zero) are both naturally identified with <math>\ell^1</math>. | ||
By the [[Riesz representation theorem]], the continuous dual of a Hilbert space is again a Hilbert space which is [[antiisomorphic|anti-isomorphic]] to the original space. | By the [[Riesz representation theorem]], the continuous dual of a Hilbert space is again a Hilbert space which is [[antiisomorphic|anti-isomorphic]] to the original space. | ||
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{{See also|Transpose of a linear map|Dual system#Transposes}} | {{See also|Transpose of a linear map|Dual system#Transposes}} | ||
If | If <math>T: V \to W</math> is a continuous linear map between two topological vector spaces, then the (continuous) transpose <math>T': W' \to V'</math> is defined by the same formula as before: | ||
<math display = "block">T'(\varphi) = \varphi \circ T, \quad \varphi \in W'.</math> | |||
The resulting functional | The resulting functional <math>T'(\varphi)</math> is in <math>V'</math>. The assignment <math>T \to T'</math> produces a linear map between the space of continuous linear maps from <math>V</math> to <math>W</math> and the space of linear maps from <math>W'</math> to <math>V'</math>. | ||
When | When <math>T</math> and <math>U</math> are composable continuous linear maps, then | ||
<math display = "block">(U \circ T)' = T' \circ U'.</math> | |||
When | When <math>V</math> and <math>W</math> are normed spaces, the norm of the transpose in <math>L(W', V')</math> is equal to that of <math>T</math> in <math>L(V, W)</math>. | ||
Several properties of transposition depend upon the [[Hahn–Banach theorem]]. | Several properties of transposition depend upon the [[Hahn–Banach theorem]]. | ||
For example, the bounded linear map | For example, the bounded linear map <math>T</math> has dense range [[if and only if]] the transpose <math>T'</math> is injective. | ||
When | When <math>T</math> is a [[Compact operator|compact]] linear map between two Banach spaces <math>V</math> and <math>W</math>, then the transpose <math>T'</math> is compact. | ||
This can be proved using the [[Arzelà–Ascoli theorem]]. | This can be proved using the [[Arzelà–Ascoli theorem]]. | ||
When | When <math>V</math> is a Hilbert space, there is an antilinear isomorphism <math>i_V</math> from <math>V</math> onto its continuous dual <math>V'</math>. | ||
For every bounded linear map | For every bounded linear map <math>T</math> on <math>V</math>, the transpose and the [[Hermitian adjoint|adjoint]] operators are linked by | ||
<math display = "block">i_V \circ T^* = T' \circ i_V.</math> | |||
When | When <math>T</math> is a continuous linear map between two topological vector spaces <math>V</math> and <math>W</math>, then the transpose <math>T'</math> is continuous when <math>W'</math> and <math>V'</math> are equipped with "compatible" topologies: for example, when for <math>X = V</math> and <math>X = W</math>, both duals <math>X'</math> have the [[Strong topology (polar topology)|strong topology]] <math>\beta(X', X)</math> of uniform convergence on bounded sets of <math>X</math>, or both have the weak-∗ topology <math>\sigma(X', X)</math> of pointwise convergence on <math>X</math>. | ||
The transpose | The transpose <math>T'</math> is continuous from <math>\beta(W', W)</math> to <math>\beta(V', V)</math>, or from <math>\sigma(W', W)</math> to <math>\sigma(V', V)</math>. | ||
=== Annihilators === | === Annihilators === | ||
Assume that | Assume that <math>W</math> is a closed linear subspace of a normed space <math>V</math>, and consider the annihilator of <math>W</math> in <math>V'</math>, | ||
<math display = "block">W^\perp = \{ \varphi \in V' : W \subseteq \ker \varphi\}.</math> | |||
Then, the dual of the quotient | Then, the dual of the quotient <math>V/W</math> can be identified with <math>W^\perp</math>, and the dual of <math>W</math> can be identified with the quotient <math>V'/{W^\perp}</math>.<ref>{{harvnb|Rudin|1991|loc=chapter 4}}</ref> | ||
Indeed, let | Indeed, let <math>P</math> denote the canonical [[surjection]] from <math>V</math> onto the quotient <math>V/W</math>. Then the transpose <math>P'</math> is an isometric isomorphism from <math>(V/W)'</math> into <math>V'</math>, with range equal to <math>W^\perp</math>. | ||
If | If <math>j</math> denotes the injection map from <math>W</math> into <math>V</math>, then the kernel of the transpose <math>j'</math> is the annihilator of <math>W</math>: | ||
<math display = "block">\ker (j') = W^\perp</math> | |||
and it follows from the [[Hahn–Banach theorem]] that | and it follows from the [[Hahn–Banach theorem]] that <math>j'</math> induces an isometric isomorphism | ||
<math>V/W^\perp</math>. | |||
=== Further properties === | === Further properties === | ||
If the dual of a normed space | If the dual of a normed space <math>V</math> is [[separable space|separable]], then so is the space <math>V</math> itself. | ||
The converse is not true: for example, the space | The converse is not true: for example, the space <math>l^1</math> is separable, but its dual <math>\ell^\infty</math> is not. | ||
=== Double dual === | === Double dual === | ||
[[File:Double dual nature.svg|thumbnail|This is a [[natural transformation]] of vector addition from a vector space to its double dual. | [[File:Double dual nature.svg|thumbnail|This is a [[natural transformation]] of vector addition from a vector space to its double dual. <math>\langle x_1, x_2\rangle</math> denotes the [[ordered pair]] of two vectors. The addition + sends <math>x_1</math> and <math>x_2</math> to <math>x_1 + x_2</math>. | ||
The addition + | The addition <math>+'</math> induced by the transformation can be defined as ''<math>[\Psi(x_1) +' \Psi(x_2)](\varphi) = \varphi(x_1 + x_2) = \varphi(x)</math>'' for any ''<math>\varphi</math>'' in the dual space.]] | ||
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator | In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator <math>\Psi: V \to V''</math> from a normed space <math>V</math> into its continuous double dual <math>V'</math>, defined by | ||
<math display = "block"> \Psi(x)(\varphi) = \varphi(x), \quad x \in V, \ \varphi \in V' .</math> | |||
As a consequence of the [[Hahn–Banach theorem]], this map is in fact an [[isometry]], meaning | As a consequence of the [[Hahn–Banach theorem]], this map is in fact an [[isometry]], meaning <math>\| \Psi(x) \| = \| x \|</math> for all <math>x \in V</math>. | ||
Normed spaces for which the map | Normed spaces for which the map <math>\Psi</math> is a [[bijection]] are called [[reflexive space|reflexive]]. | ||
When | When <math>V</math> is a [[topological vector space]] then <math>\Psi</math>(<math>x</math>) can still be defined by the same formula, for every <math>x \in V</math>, however several difficulties arise. | ||
First, when | First, when <math>V</math> is not [[Locally convex topological vector space|locally convex]], the continuous dual may be equal to { 0 } and the map <math>\Psi</math> trivial. | ||
However, if | However, if <math>V</math> is [[Hausdorff space|Hausdorff]] and locally convex, the map <math>\Psi</math> is injective from <math>V</math> to the algebraic dual <math>V^*</math> of the continuous dual, again as a consequence of the Hahn–Banach theorem.<ref group=nb>If <math>V</math> is locally convex but not Hausdorff, the [[kernel (algebra)|kernel]] of <math>\Psi</math> is the smallest closed subspace containing {0}.</ref> | ||
Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual | Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual <math>V'</math>, so that the continuous double dual <math>V''</math> is not uniquely defined as a set. Saying that <math>\Psi</math> maps from <math>V</math> to <math>V''</math>, or in other words, that <math>\Psi(x)</math> is continuous on <math>V'</math> for every <math>x \in V</math>, is a reasonable minimal requirement on the topology of <math>V'</math>, namely that the evaluation mappings | ||
<math display = "block"> \varphi \in V' \mapsto \varphi(x), \quad x \in V , </math> | |||
be continuous for the chosen topology on | be continuous for the chosen topology on <math>V'</math>. Further, there is still a choice of a topology on <math>V''</math>, and continuity of <math>\Psi</math> depends upon this choice. | ||
As a consequence, defining [[Reflexive space#Locally convex spaces|reflexivity]] in this framework is more involved than in the normed case. | As a consequence, defining [[Reflexive space#Locally convex spaces|reflexivity]] in this framework is more involved than in the normed case. | ||
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| publisher = Springer-Verlag | | publisher = Springer-Verlag | ||
}} | }} | ||
* {{Cite book|last=Halmos|first=Paul Richard|title=Finite-Dimensional Vector Spaces|publisher= [[Springer Science+Business Media|Springer]]|year=1974|isbn=0-387-90093-4|edition= | * {{Cite book|last=Halmos|first=Paul Richard|title=Finite-Dimensional Vector Spaces|publisher= [[Springer Science+Business Media|Springer]]|year=1974|isbn=0-387-90093-4|edition=1st|author-link=Paul Halmos|orig-year=1958}} | ||
* {{Cite book|last1=Katznelson|first1=Yitzhak|title=A (Terse) Introduction to Linear Algebra|last2=Katznelson|first2=Yonatan R.|publisher=[[American Mathematical Society]]|year=2008|isbn=978-0-8218-4419-9|author-link1=Yitzhak Katznelson}} | * {{Cite book|last1=Katznelson|first1=Yitzhak|title=A (Terse) Introduction to Linear Algebra|last2=Katznelson|first2=Yonatan R.|publisher=[[American Mathematical Society]]|year=2008|isbn=978-0-8218-4419-9|author-link1=Yitzhak Katznelson}} | ||
* {{Lang Algebra|edition=3r}} | * {{Lang Algebra|edition=3r}} | ||
* {{Cite book|last=Tu|first=Loring W.|title=An Introduction to Manifolds|publisher=[[Springer Science+Business Media|Springer]]|year=2011|isbn=978-1-4419-7400-6|edition=2nd|pages=|author-link=Loring W. Tu}} | * {{Cite book|last=Tu|first=Loring W.|title=An Introduction to Manifolds|title-link=An Introduction to Manifolds|publisher=[[Springer Science+Business Media|Springer]]|year=2011|isbn=978-1-4419-7400-6|edition=2nd|pages=|author-link=Loring W. Tu}} | ||
* {{Cite book | last1=Mac Lane | first1=Saunders | author-link1=Saunders Mac Lane | last2=Birkhoff | first2=Garrett | author-link2=Garrett Birkhoff | title=Algebra|edition=3rd | publisher=AMS Chelsea Publishing | year=1999 | isbn=0-8218-1646-2}}. | * {{Cite book | last1=Mac Lane | first1=Saunders | author-link1=Saunders Mac Lane | last2=Birkhoff | first2=Garrett | author-link2=Garrett Birkhoff | title=Algebra|edition=3rd | publisher=AMS Chelsea Publishing | year=1999 | isbn=0-8218-1646-2}}. | ||
* {{Cite book | * {{Cite book | ||
| Line 457: | Line 460: | ||
[[Category:Functional analysis]] | [[Category:Functional analysis]] | ||
[[Category:Linear algebra]] | [[Category:Linear algebra]] | ||
[[Category:Duality | [[Category:Duality (mathematics)|Space]] | ||
[[Category:Linear functionals]] | [[Category:Linear functionals]] | ||
Latest revision as of 21:59, 26 March 2026
In mathematics, any vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants.
The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.
Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.
Early terms for dual include polarer Raum [Hahn 1927], espace conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938.[1]
Algebraic dual space
Given any vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} over a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , the (algebraic) dual space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{*}} [2] (alternatively denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{\lor}} [3] or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} [4][5])[nb 1] is defined as the set of all linear maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi: V \to F} (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hom (V, F)} .[3] The dual space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} itself becomes a vector space over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} when equipped with an addition and scalar multiplication satisfying: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} (\varphi + \psi)(x) &= \varphi(x) + \psi(x) \\ (a \varphi)(x) &= a \left(\varphi(x)\right) \end{align}} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi, \psi \in V^*} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in V} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in F} . For example, if we express the vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Reals^2} as the set of vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\mathbf{i} + s\mathbf{j}} (where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} are real numbers), the function
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(r\mathbf{i} + s\mathbf{j}) = r + s}
is an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Reals^2)^*} , since it is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Reals} -linear and maps vectors in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Reals^2} to elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Reals} .
Elements of the algebraic dual space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} are sometimes called covectors, one-forms, or linear forms.
The pairing of a functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} in the dual space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} and an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is sometimes denoted by a bracket: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi (x) = [x, \varphi]} [6] or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi (x) = \langle x, \varphi \rangle} .[7] This pairing defines a nondegenerate bilinear mapping[nb 2] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \cdot, \cdot \rangle : V \times V^* \to F} called the natural pairing.
Dual set
Given a vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and a basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} on that space, one can define a linearly independent set in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} called the dual set. Each vector in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} corresponds to a unique vector in the dual set. This correspondence yields an injection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \to V^*} .
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite-dimensional, the dual set is a basis, called the dual basis, and the injection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \to V^*} is an isomorphism.
Finite-dimensional case
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite-dimensional and has a basis, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{e}_1,\dots,\mathbf{e}_n\}} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , the dual basis is a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{e}^1,\dots,\mathbf{e}^n\}} of linear functionals on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , defined by the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^i(c^1 \mathbf{e}_1+\cdots+c^n\mathbf{e}_n) = c^i, \quad i=1,\ldots,n } for any choice of coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^i\in F} . In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^i(\mathbf{e}_j) = \delta^{i}_{j} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta^{i}_{j}} is the Kronecker delta symbol. This property is referred to as the bi-orthogonality property.
Proof
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Consider Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{e}_1,\dots,\mathbf{e}_n\}} the basis of V. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{e}^1,\dots,\mathbf{e}^n\}} be defined as the following: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^i(c^1 \mathbf{e}_1+\cdots+c^n\mathbf{e}_n) = c^i, \quad i=1,\ldots,n } . These are a basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} because:
Hence, it is a basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} . |
For example, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^2} , let its basis be chosen as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{e}_1=(1/2,1/2),\mathbf{e}_2=(0,1)\}} . The basis vectors are not orthogonal to each other. Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^2} are one-forms (functions that map a vector to a scalar) such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^1(\mathbf{e}_1)=1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^1(\mathbf{e}_2)=0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^2(\mathbf{e}_1)=0} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^2(\mathbf{e}_2)=1} . (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} e^{11} & e^{12} \\ e^{21} & e^{22} \end{bmatrix} \begin{bmatrix} e_{11} & e_{21} \\ e_{12} & e_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. } Solving for the unknown values in the first matrix shows the dual basis to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{e}^1=(2,0),\mathbf{e}^2=(-1,1)\}} . Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^2} are functionals, they can be rewritten as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^1(x,y)=2x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^2(x,y)=-x+y} .
In general, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} , if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=[\mathbf{e}_1|\cdots|\mathbf{e}_n]} is a matrix whose columns are the basis vectors and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{E}=[\mathbf{e}^1|\cdots|\mathbf{e}^n]} is a matrix whose columns are the dual basis vectors, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{E}^\textrm{T}\cdot E = I_n,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_n} is the identity matrix of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . The biorthogonality property of these two basis sets allows any point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}\in V} to be represented as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} = \sum_i \langle\mathbf{x},\mathbf{e}^i \rangle \mathbf{e}_i = \sum_i \langle \mathbf{x}, \mathbf{e}_i \rangle \mathbf{e}^i,}
even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \cdot, \cdot \rangle} and the corresponding duality pairing are introduced, as described below in § Bilinear products and dual spaces.
In particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} can be interpreted as the space of columns of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} real numbers, its dual space is typically written as the space of rows of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} real numbers. Such a row acts on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} as a linear functional by ordinary matrix multiplication. This is because a functional maps every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} into a real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} . Then, seeing this functional as a matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} as an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times 1} matrix, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\times 1} matrix (trivially, a real number) respectively, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Mx=y} then, by dimension reasons, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} must be a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\times n} matrix; that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} must be a row vector.
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} consists of the space of geometrical vectors in the plane, then the level curves of an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} form a family of parallel lines in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. So an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, it suffices to determine which of the lines the vector lies on. Informally, this "counts" how many lines the vector crosses. More generally, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a vector space of any dimension, then the level sets of a linear functional in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} are parallel hyperplanes in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.[8]
Infinite-dimensional case
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is not finite-dimensional but has a basis[nb 3] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_\alpha} indexed by an infinite set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , then the same construction as in the finite-dimensional case yields linearly independent elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}^\alpha} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\in A} ) of the dual space, but they will not form a basis.
For instance, consider the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^\infty} , whose elements are those sequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \N} . For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in \N} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_i} is the sequence consisting of all zeroes except in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} -th position, which is 1. The dual space of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^\infty} is (isomorphic to) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^\N} , the space of all sequences of real numbers: each real sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} defines a function where the element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^\infty} is sent to the number
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_n a_nx_n,}
which is a finite sum because there are only finitely many nonzero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n} . The dimension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^\infty} is countably infinite, whereas Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^\N} does not have a countable basis.
This observation generalizes to any[nb 3] infinite-dimensional vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} over any field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} : a choice of basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf{e}_\alpha:\alpha\in A\}} identifies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} with the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (F^A)_0} of functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:A\to F} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\alpha=f(\alpha)} is nonzero for only finitely many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\in A} , where such a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is identified with the vector
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\alpha\in A} f_\alpha\mathbf{e}_\alpha}
in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} (the sum is finite by the assumption on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , and any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\in V} may be written uniquely in this way by the definition of the basis).
The dual space of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} may then be identified with the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^A} of all functions from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} : a linear functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is uniquely determined by the values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_\alpha=T(\mathbf{e}_\alpha)} it takes on the basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , and any function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta:A\to F} (with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta(\alpha)=\theta_\alpha} ) defines a linear functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\left (\sum_{\alpha\in A} f_\alpha \mathbf{e}_\alpha\right) = \sum_{\alpha \in A} f_\alpha T(e_\alpha) = \sum_{\alpha\in A} f_\alpha \theta_\alpha.}
Again, the sum is finite because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\alpha} is nonzero for only finitely many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} .
The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (F^A)_0} may be identified (essentially by definition) with the direct sum of infinitely many copies of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} (viewed as a 1-dimensional vector space over itself) indexed by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , i.e. there are linear isomorphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\cong (F^A)_0\cong\bigoplus_{\alpha\in A} F.}
On the other hand, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^A} is (again by definition), the direct product of infinitely many copies of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} indexed by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , and so the identification Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^* \cong \left (\bigoplus_{\alpha\in A}F\right )^* \cong \prod_{\alpha\in A}F^* \cong \prod_{\alpha\in A}F \cong F^A} is a special case of a general result relating direct sums (of modules) to direct products.
If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional.
The proof of this inequality between dimensions results from the following.
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is an infinite-dimensional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} -vector space, the arithmetical properties of cardinal numbers implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{dim}(V)=|A|<|F|^{|A|}=|V^\ast|=\mathrm{max}(|\mathrm{dim}(V^\ast)|, |F|),} where cardinalities are denoted as absolute values. For proving that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{dim}(V)< \mathrm{dim}(V^*),} it suffices to prove that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F|\le |\mathrm{dim}(V^\ast)|,} which can be done with an argument similar to Cantor's diagonal argument.[9] The exact dimension of the dual is given by the Erdős–Kaplansky theorem.
Bilinear products and dual spaces
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite-dimensional, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} . But there is in general no natural isomorphism between these two spaces.[10] Any bilinear form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \cdot, \cdot \rangle} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} gives a mapping of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} into its dual space via
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\mapsto \langle v, \cdot\rangle}
where the right hand side is defined as the functional on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} taking each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w \in V} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle v, w \rangle} . In other words, the bilinear form determines a linear mapping
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{\langle\cdot,\cdot\rangle} : V\to V^*}
defined by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\Phi_{\langle\cdot,\cdot\rangle}(v), w\right] = \langle v, w\rangle.}
If the bilinear form is nondegenerate, then this is an isomorphism onto a subspace of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite-dimensional, then this is an isomorphism onto all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} . Conversely, any isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} to a subspace of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} (resp., all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite dimensional) defines a unique nondegenerate bilinear form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \cdot, \cdot \rangle_{\Phi} } on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle v, w \rangle_\Phi = (\Phi (v))(w) = [\Phi (v), w].\,}
Thus there is a one-to-one correspondence between isomorphisms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} to a subspace of (resp., all of) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} and nondegenerate bilinear forms on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} .
If the vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is over the complex field, then sometimes it is more natural to consider sesquilinear forms instead of bilinear forms. In that case, a given sesquilinear form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \cdot, \cdot \rangle} determines an isomorphism of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} with the complex conjugate of the dual space
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{\langle \cdot, \cdot \rangle} : V\to \overline{V^*}. } The conjugate of the dual space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{V^*}} can be identified with the set of all additive complex-valued functionals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: V \to \Complex} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\alpha v) = \overline{\alpha}f(v). }
Injection into the double-dual
There is a natural homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} into the double dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{**}=\hom (V^*, F)} , defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Psi(v))(\varphi)=\varphi(v)} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\in V, \varphi\in V^*} . In other words, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ev}_v:V^*\to F} is the evaluation map defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi \mapsto \varphi(v)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi: V \to V^{**}} is defined as the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\mapsto\mathrm{ev}_v} . This map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} is always injective;[nb 3] and it is always an isomorphism if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite-dimensional.[11] Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural isomorphism. Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.
Transpose of a linear map
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: V \to W} is a linear map, then the transpose (or dual) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^*: W^* \to V^*} is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^*(\varphi) = \varphi \circ f \, } for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi \in W^*} . The resulting functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^* (\varphi)} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} is called the pullback of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} .
The following identity holds for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi \in W^*} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \in V} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [f^*(\varphi),\, v] = [\varphi,\, f(v)], } where the bracket Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\cdot,\cdot]} on the left is the natural pairing of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} with its dual space, and that on the right is the natural pairing of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} with its dual. This identity characterizes the transpose,[12] and is formally similar to the definition of the adjoint.
The assignment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \mapsto f^*} produces an injective linear map between the space of linear operators from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} and the space of linear operators from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W^*} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} ; this homomorphism is an isomorphism if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is finite-dimensional. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = W} then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {(fg)}^* = g^*f^*} . In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} to itself. It is possible to identify Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{**}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} using the natural injection into the double dual.
If the linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is represented by the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} with respect to two bases of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^*} is represented by the transpose matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^T} with respect to the dual bases of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W^*} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} , hence the name. Alternatively, as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is represented by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} acting on the left on column vectors, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^*} is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Reals^n} , which identifies the space of column vectors with the dual space of row vectors.
Quotient spaces and annihilators
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} be a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . The annihilator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} , denoted here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^0} , is the collection of linear functionals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\in V^*} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [f,s]=0} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s\in S} . That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^0} consists of all linear functionals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:V\to F} such that the restriction to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} vanishes: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f|_S = 0} . Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement.
The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 0 \}^0 = V^*} , and the annihilator of the whole space is just the zero covector: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^0 = \{ 0 \} \subseteq V^*} . Furthermore, the assignment of an annihilator to a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} reverses inclusions, so that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 0 \} \subseteq S\subseteq T\subseteq V} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ 0 \} \subseteq T^0 \subseteq S^0 \subseteq V^* . }
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} are two subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^0 + B^0 \subseteq (A \cap B)^0 . } If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A_i)_{i\in I}} is any family of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} indexed by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} belonging to some index set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \bigcup_{i\in I} A_i \right)^0 = \bigcap_{i\in I} A_i^0 . } In particular if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} are subspaces of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A + B)^0 = A^0 \cap B^0 } and[nb 3] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \cap B)^0 = A^0 + B^0 . }
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite-dimensional and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is a vector subspace, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W^{00} = W } after identifying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} with its image in the second dual space under the double duality isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\approx V^{**}} . In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space.
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is a subspace of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} then the quotient space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V/W} is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:V\to F} factors through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V/W} if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is in the kernel of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . There is thus an isomorphism
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (V/W)^* \cong W^0 .}
As a particular consequence, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a direct sum of two subspaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} is a direct sum of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^0} .
Dimensional analysis
The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \in V} can be paired with a covector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi \in V^*} by the natural pairing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x, \varphi \rangle := \varphi (x) \in F} to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to reducing a fraction. Thus while the direct sum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \oplus V^*} is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} -dimensional space (if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -dimensional), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} behaves as an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -n} -dimensional space, in the sense that its dimensions can be canceled against the dimensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . This is formalized by tensor contraction.
This arises in physics via dimensional analysis, where the dual space has inverse units.[13] Under the natural pairing, these units cancel, and the resulting scalar value is dimensionless, as expected. For example, in (continuous) Fourier analysis, or more broadly time–frequency analysis:[nb 4] given a one-dimensional vector space with a unit of time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , the dual space has units of frequency: occurrences per unit of time (units of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/t} ). For example, if time is measured in seconds, the corresponding dual unit is the inverse second: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3s \cdot 2s^{-1} = 6} . Similarly, if the primal space measures length, the dual space measures inverse length.
Continuous dual space
When dealing with topological vector spaces, the continuous linear functionals from the space into the base field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F} = \Complex} (or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} ) are particularly important. This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} , denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} . For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps. Nevertheless, in the theory of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space".
For a topological vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} its continuous dual space,[14] or topological dual space,[15] or just dual space[14][15][16][17] (in the sense of the theory of topological vector spaces) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is defined as the space of all continuous linear functionals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi:V\to{\mathbb F}} .
Important examples for continuous dual spaces are the space of compactly supported test functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{D}} and its dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{D}',} the space of arbitrary distributions (generalized functions); the space of arbitrary test functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}} and its dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}',} the space of compactly supported distributions; and the space of rapidly decreasing test functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S},} the Schwartz space, and its dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}',} the space of tempered distributions (slowly growing distributions) in the theory of generalized functions.
Properties
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a Hausdorff topological vector space (TVS), then the continuous dual space of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is identical to the continuous dual space of the completion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .[1]
Topologies on the dual
There is a standard construction for introducing a topology on the continuous dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} of a topological vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . Fix a collection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} of bounded subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . This gives the topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} of uniform convergence on sets from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A},} or what is the same thing, the topology generated by seminorms of the form
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\varphi\|_A = \sup_{x\in A} |\varphi(x)|,}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} is a continuous linear functional on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} runs over the class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}.}
This means that a net of functionals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_i} tends to a functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} if and only if
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{ for all } A\in\mathcal{A}\qquad \|\varphi_i-\varphi\|_A = \sup_{x\in A} |\varphi_i(x)-\varphi(x)|\underset{i\to\infty}{\longrightarrow} 0. }
Usually (but not necessarily) the class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} is supposed to satisfy the following conditions:
- Each point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} belongs to some set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in\mathcal{A}} .
- Each two sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in \mathcal{A}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B \in \mathcal{A}} are contained in some set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \in \mathcal{A}} .
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} is closed under the operation of multiplication by scalars.
If these requirements are fulfilled then the corresponding topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is Hausdorff and the sets
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_A ~=~ \left \{ \varphi \in V' ~:~ \quad \|\varphi\|_A < 1 \right \},\qquad \text{ for } A \in \mathcal{A}}
form its local base.
Here are the three most important special cases.
- The strong topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is the topology of uniform convergence on bounded subsets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} (so here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} can be chosen as the class of all bounded subsets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} ).
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a normed vector space (for example, a Banach space or a Hilbert space) then the strong topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is normed (in fact a Banach space if the field of scalars is complete), with the norm Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\varphi\| = \sup_{\|x\| \le 1 } |\varphi(x)|.}
- The stereotype topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is the topology of uniform convergence on totally bounded sets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} (so here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} can be chosen as the class of all totally bounded subsets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} ).
- The weak topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is the topology of uniform convergence on finite subsets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} (so here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} can be chosen as the class of all finite subsets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} ).
Each of these three choices of topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} leads to a variant of reflexivity property for topological vector spaces:
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is endowed with the strong topology, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called reflexive.[18]
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of stereotype spaces: the spaces reflexive in this sense are called stereotype.
- If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} is endowed with the weak topology, then the corresponding reflexivity is presented in the theory of dual pairs:[19] the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.[20]
Examples
Let 1 < Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} < ∞ be a real number and consider the Banach space ℓ p of all sequences Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{a} = a_n} for which
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\mathbf{a}\|_p = \left ( \sum_{n=0}^\infty |a_n|^p \right) ^{\frac{1}{p}} < \infty.}
Define the number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/p + 1/q = 1} . Then the continuous dual of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^p} is naturally identified with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^q} : given an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi \in (\ell^p)'} , the corresponding element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^q} is the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\varphi(\mathbf {e}_n))} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf {e}_n} denotes the sequence whose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -th term is 1 and all others are zero. Conversely, given an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{a} = (a_n) \in \ell^q} , the corresponding continuous linear functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^p} is defined by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi (\mathbf{b}) = \sum_n a_n b_n}
for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{b} = (b_n) \in \ell^p} (see Hölder's inequality).
In a similar manner, the continuous dual of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^1} is naturally identified with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^\infty} (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} (consisting of all convergent sequences, with the supremum norm) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_0} (the sequences converging to zero) are both naturally identified with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^1} .
By the Riesz representation theorem, the continuous dual of a Hilbert space is again a Hilbert space which is anti-isomorphic to the original space. This gives rise to the bra–ket notation used by physicists in the mathematical formulation of quantum mechanics.
By the Riesz–Markov–Kakutani representation theorem, the continuous dual of certain spaces of continuous functions can be described using measures.
Transpose of a continuous linear map
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T: V \to W} is a continuous linear map between two topological vector spaces, then the (continuous) transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T': W' \to V'} is defined by the same formula as before:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T'(\varphi) = \varphi \circ T, \quad \varphi \in W'.}
The resulting functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T'(\varphi)} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} . The assignment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T \to T'} produces a linear map between the space of continuous linear maps from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} and the space of linear maps from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W'} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} . When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} are composable continuous linear maps, then
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (U \circ T)' = T' \circ U'.}
When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} are normed spaces, the norm of the transpose in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(W', V')} is equal to that of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(V, W)} . Several properties of transposition depend upon the Hahn–Banach theorem. For example, the bounded linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} has dense range if and only if the transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T'} is injective.
When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is a compact linear map between two Banach spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} , then the transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T'} is compact. This can be proved using the Arzelà–Ascoli theorem.
When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a Hilbert space, there is an antilinear isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_V} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} onto its continuous dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} . For every bounded linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , the transpose and the adjoint operators are linked by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_V \circ T^* = T' \circ i_V.}
When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is a continuous linear map between two topological vector spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} , then the transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T'} is continuous when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W'} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} are equipped with "compatible" topologies: for example, when for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = W} , both duals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X'} have the strong topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(X', X)} of uniform convergence on bounded sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , or both have the weak-∗ topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(X', X)} of pointwise convergence on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . The transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T'} is continuous from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(W', W)} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(V', V)} , or from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(W', W)} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(V', V)} .
Annihilators
Assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is a closed linear subspace of a normed space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , and consider the annihilator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W^\perp = \{ \varphi \in V' : W \subseteq \ker \varphi\}.}
Then, the dual of the quotient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V/W} can be identified with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W^\perp} , and the dual of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} can be identified with the quotient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'/{W^\perp}} .[21] Indeed, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} denote the canonical surjection from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} onto the quotient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V/W} . Then the transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'} is an isometric isomorphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (V/W)'} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} , with range equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W^\perp} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} denotes the injection map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , then the kernel of the transpose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j'} is the annihilator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ker (j') = W^\perp} and it follows from the Hahn–Banach theorem that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j'} induces an isometric isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V/W^\perp} .
Further properties
If the dual of a normed space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is separable, then so is the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} itself. The converse is not true: for example, the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l^1} is separable, but its dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^\infty} is not.
Double dual
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi: V \to V''} from a normed space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} into its continuous double dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} , defined by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(x)(\varphi) = \varphi(x), \quad x \in V, \ \varphi \in V' .}
As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \Psi(x) \| = \| x \|} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in V} . Normed spaces for which the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} is a bijection are called reflexive.
When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a topological vector space then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} ) can still be defined by the same formula, for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in V} , however several difficulties arise. First, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is not locally convex, the continuous dual may be equal to { 0 } and the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} trivial. However, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is Hausdorff and locally convex, the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} is injective from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} to the algebraic dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} of the continuous dual, again as a consequence of the Hahn–Banach theorem.[nb 5]
Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} , so that the continuous double dual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V''} is not uniquely defined as a set. Saying that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} maps from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V''} , or in other words, that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(x)} is continuous on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in V} , is a reasonable minimal requirement on the topology of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} , namely that the evaluation mappings
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi \in V' \mapsto \varphi(x), \quad x \in V , }
be continuous for the chosen topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} . Further, there is still a choice of a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V''} , and continuity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case.
See also
- Covariance and contravariance of vectors
- Dual module
- Dual norm
- Duality (mathematics)
- Duality (projective geometry)
- Pontryagin duality
- Reciprocal lattice – dual space basis, in crystallography
Notes
- ↑ For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{\lor}} used in this way, see An Introduction to Manifolds (Tu 2011, p. 19). This notation is sometimes used when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\cdot)^*} is reserved for some other meaning. For instance, in the above text, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^*} is frequently used to denote the codifferential of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^* \omega} represents the pullback of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} . Halmos (1974, p. 20) uses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} to denote the algebraic dual of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . However, other authors use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V'} for the continuous dual, while reserving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^*} for the algebraic dual (Trèves 2006, p. 35).
- ↑ In many areas, such as quantum mechanics, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \cdot, \cdot \rangle} is reserved for a sesquilinear form defined on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \times V} .
- ↑ 3.0 3.1 3.2 3.3 Several assertions in this article require the axiom of choice for their justification. The axiom of choice is needed to show that an arbitrary vector space has a basis: in particular it is needed to show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^\N} has a basis. It is also needed to show that the dual of an infinite-dimensional vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is nonzero, and hence that the natural map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} to its double dual is injective.
- ↑ To be precise, continuous Fourier analysis studies the space of functionals with domain a vector space and the space of functionals on the dual vector space.
- ↑ If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is locally convex but not Hausdorff, the kernel of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} is the smallest closed subspace containing {0}.
References
- ↑ 1.0 1.1 Narici & Beckenstein 2011, pp. 225–273.
- ↑ Katznelson & Katznelson (2008) p. 37, §2.1.3
- ↑ 3.0 3.1 Tu (2011) p. 19, §3.1
- ↑ Axler (2015) p. 101, §3.94
- ↑ Halmos (1974) p. 20, §13
- ↑ Halmos (1974) p. 21, §14
- ↑ Misner, Thorne & Wheeler 1973
- ↑ Misner, Thorne & Wheeler 1973, §2.5
- ↑ Nicolas Bourbaki (1974). Hermann (ed.). Elements of mathematics: Algebra I, Chapters 1 - 3. Addison-Wesley Publishing Company. p. 400. ISBN 0201006391.
- ↑ Mac Lane & Birkhoff 1999, §VI.4
- ↑ Halmos (1974) pp. 25, 28
- ↑ Halmos (1974) §44
- ↑ Tao, Terence (2012-12-29). "A mathematical formalisation of dimensional analysis".
Similarly, one can define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{T^{-1}}} as the dual space to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^T} ...
- ↑ 14.0 14.1 Robertson & Robertson 1964, II.2
- ↑ 15.0 15.1 Schaefer 1966, II.4
- ↑ Rudin 1973, 3.1
- ↑ Bourbaki 2003, II.42
- ↑ Schaefer 1966, IV.5.5
- ↑ Schaefer 1966, IV.1
- ↑ Schaefer 1966, IV.1.2
- ↑ Rudin 1991, chapter 4
Bibliography
- Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.
- Bourbaki, Nicolas (1989). Elements of mathematics, Algebra I. Springer-Verlag. ISBN 3-540-64243-9.
- Bourbaki, Nicolas (2003). Elements of mathematics, Topological vector spaces. Springer-Verlag.
- Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces (1st ed.). Springer. ISBN 0-387-90093-4.
- Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
- Template:Lang Algebra
- Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). Springer. ISBN 978-1-4419-7400-6.
- Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). AMS Chelsea Publishing. ISBN 0-8218-1646-2..
- Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973). Gravitation. W. H. Freeman. ISBN 0-7167-0344-0.
- Template:Narici Beckenstein Topological Vector Spaces
- Template:Rudin Walter Functional Analysis
- Template:Rudin Walter Functional Analysis
- Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press.
- Schaefer, Helmut H. (1966). Topological vector spaces. New York: The Macmillan Company.
- Template:Schaefer Wolff Topological Vector Spaces
- Template:Trèves François Topological vector spaces, distributions and kernels
External links
Template:Linear algebra Template:Functional analysis Template:Duality and spaces of linear maps