Inner product space: Difference between revisions
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{{short description|Vector space with generalized dot product}} | {{short description|Vector space with generalized dot product}} | ||
{{redirect|Inner product|the inner product of coordinate vectors|Dot product}} | {{redirect|Inner product|the inner product of coordinate vectors|Dot product}} | ||
[[File:Inner-product-angle. | [[File:Inner-product-angle.svg|thumb|300px|Geometric interpretation of the angle between two vectors defined using an inner product]] | ||
[[File:Product Spaces Drawing (1).png|alt=Scalar product spaces, inner product spaces, Hermitian product spaces.|thumb|300px|Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.]] | [[File:Product Spaces Drawing (1).png|alt=Scalar product spaces, inner product spaces, Hermitian product spaces.|thumb|300px|Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.]] | ||
In [[mathematics]], an '''inner product space''' | In [[mathematics]], an '''inner product space'''{{refn|group="Note"|Also called, rarely, a '''Hausdorff pre-Hilbert space''',{{sfn|Trèves|2006|pp=112-125}}{{sfn|Schaefer|Wolff|1999|pp=40-45}} titled after [[Hausdorff space]]s and [[Hilbert space]]s.}} is a [[real vector space|real]] or [[complex vector space]] endowed with an [[operation (mathematics)|operation]] called an '''inner product'''. The inner product of two vectors in the space is a [[Scalar (mathematics)|scalar]], often denoted with [[angle brackets]] such as in <math>\langle a, b \rangle</math>. Inner products allow formal definitions of intuitive geometric notions, such as lengths, [[angle]]s, and [[orthogonality]] (zero inner product) of vectors. Inner product spaces generalize [[Euclidean vector space]]s, in which the inner product is the [[dot product]] or ''scalar product'' of [[Cartesian coordinates]]. Inner product spaces of infinite [[Dimension (vector space)|dimensions]] are widely used in [[functional analysis]]. Inner product spaces over the [[Field (mathematics)|field]] of [[complex number]]s are sometimes referred to as '''unitary spaces'''. The first usage of the concept of a vector space with an inner product is due to [[Giuseppe Peano]], in 1898.<ref>{{cite journal|last1=Moore|first1=Gregory H.|title=The axiomatization of linear algebra: 1875-1940|journal=Historia Mathematica|date=1995|volume=22|issue=3|pages=262–303|doi=10.1006/hmat.1995.1025|doi-access=free}}</ref> | ||
An inner product naturally induces an associated [[Norm (mathematics)|norm]], (denoted <math>|x|</math> and <math>|y|</math> in the picture); so, every inner product space is a [[normed vector space]]. If this normed space is also [[complete metric space|complete]] (that is, a [[Banach space]]) then the inner product space is a [[Hilbert space]].{{sfn|Trèves|2006|pp=112-125}} If an inner product space {{mvar|H}} is not a Hilbert space, it can be ''extended'' by [[Complete topological vector space#Completions|completion]] to a Hilbert space <math>\overline{H}.</math> This means that <math>H</math> is a [[linear subspace]] of <math>\overline{H},</math> the inner product of <math>H</math> is the [[restriction (mathematics)|restriction]] of that of <math>\overline{H},</math> and <math>H</math> is [[Dense subset|dense]] in <math>\overline{H}</math> for the [[topology (structure)|topology]] defined by the norm.{{sfn|Trèves|2006|pp=112-125}}{{sfn|Schaefer|Wolff|1999|pp=36-72}} | An inner product naturally induces an associated [[Norm (mathematics)|norm]], (denoted <math>|x|</math> and <math>|y|</math> in the picture); so, every inner product space is a [[normed vector space]]. If this normed space is also [[complete metric space|complete]] (that is, a [[Banach space]]) then the inner product space is a [[Hilbert space]].{{sfn|Trèves|2006|pp=112-125}} If an inner product space {{mvar|H}} is not a Hilbert space, it can be ''extended'' by [[Complete topological vector space#Completions|completion]] to a Hilbert space <math>\overline{H}.</math> This means that <math>H</math> is a [[linear subspace]] of <math>\overline{H},</math> the inner product of <math>H</math> is the [[restriction (mathematics)|restriction]] of that of <math>\overline{H},</math> and <math>H</math> is [[Dense subset|dense]] in <math>\overline{H}</math> for the [[topology (structure)|topology]] defined by the norm.{{sfn|Trèves|2006|pp=112-125}}{{sfn|Schaefer|Wolff|1999|pp=36-72}} | ||
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An ''inner product'' space is a [[vector space]] {{math|''V''}} over the field {{math|''F''}} together with an ''inner product'', that is, a map | An ''inner product'' space is a [[vector space]] {{math|''V''}} over the field {{math|''F''}} together with an ''inner product'', that is, a map | ||
<math display="block"> \langle \cdot, \cdot \rangle : V \times V \to F </math> | <math display="block"> \langle \cdot \operatorname{,} \cdot \rangle : V \times V \to F </math> | ||
that satisfies the following three properties for all vectors <math>x,y,z\in V</math> and all scalars {{nowrap|<math>a,b \in F</math>.<ref name= Jain>{{cite book |title=Functional Analysis |first1=P. K. |last1=Jain |first2=Khalil |last2=Ahmad |chapter-url=https://books.google.com/books?id=yZ68h97pnAkC&pg=PA203 |page=203 |chapter=5.1 Definitions and basic properties of inner product spaces and Hilbert spaces |isbn=81-224-0801-X |year=1995 |edition=2nd |publisher=New Age International}}</ref><ref name="Prugovec̆ki">{{cite book |title=Quantum Mechanics in Hilbert Space |first=Eduard |last=Prugovečki |chapter-url=https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA18 |chapter=Definition 2.1 |pages=18ff |isbn=0-12-566060-X | year = 1981 |publisher=Academic Press |edition = 2nd}}</ref>}} | that satisfies the following three properties for all vectors <math>x,y,z\in V</math> and all scalars {{nowrap|<math>a,b \in F</math>.<ref name= Jain>{{cite book |title=Functional Analysis |first1=P. K. |last1=Jain |first2=Khalil |last2=Ahmad |chapter-url=https://books.google.com/books?id=yZ68h97pnAkC&pg=PA203 |page=203 |chapter=5.1 Definitions and basic properties of inner product spaces and Hilbert spaces |isbn=81-224-0801-X |year=1995 |edition=2nd |publisher=New Age International}}</ref><ref name="Prugovec̆ki">{{cite book |title=Quantum Mechanics in Hilbert Space |first=Eduard |last=Prugovečki |chapter-url=https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA18 |chapter=Definition 2.1 |pages=18ff |isbn=0-12-566060-X | year = 1981 |publisher=Academic Press |edition = 2nd}}</ref>}} | ||
* ''Conjugate symmetry'': <math display=block>\langle x, y \rangle = \overline{\langle y, x \rangle}.</math> As <math display="inline"> a = \overline{a} </math> [[if and only if]] <math>a</math> is real, conjugate symmetry implies that <math>\langle x, x \rangle </math> is always a real number. If {{math|''F''}} is <math>\R</math>, conjugate symmetry is just symmetry. | * ''Conjugate symmetry'': <math display=block>\langle x, y \rangle = \overline{\langle y, x \rangle}.</math> As <math display="inline"> a = \overline{a} </math> [[if and only if]] <math>a</math> is real, conjugate symmetry implies that <math>\langle x, x \rangle </math> is always a real number. If {{math|''F''}} is <math>\R</math>, conjugate symmetry is just symmetry. | ||
* [[Linear map|Linearity]] in the first argument:<ref group="Note"> | * [[Linear map|Linearity]] in the first argument:<ref group="Note">Combining the ''linearity in the first argument'' property with the ''conjugate symmetry'' property proves ''conjugate-linearity in the second argument'': <math display="inline"> \langle x,by \rangle = \langle x,y \rangle \overline{b} </math>. This is how the inner product was originally defined and is used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in the [[bra-ket]] notation of [[Paul Dirac]], where the inner product is taken to be ''linear in the second argument'' and ''conjugate-linear in the first argument''; this convention is used in many other domains such as engineering and computer science.</ref> <math display=block> | ||
\langle ax+by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle.</math> | \langle ax+by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle.</math> | ||
* [[Definite bilinear form|Positive-definiteness]]: if <math>x</math> is not zero, then <math display=block> | * [[Definite bilinear form|Positive-definiteness]]: if <math>x</math> is not zero, then <math display=block> | ||
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=== Basic properties === | === Basic properties === | ||
In the following properties, which result almost immediately from the definition of an inner product, {{math|''x'', ''y''}} and {{mvar|z}} are arbitrary vectors, and {{mvar|a}} and {{mvar|b}} are arbitrary scalars. | In the following properties, which result almost immediately from the definition of an inner product, {{math|''x'', ''y''}} and {{mvar|z}} are arbitrary vectors, and {{mvar|a}} and {{mvar|b}} are arbitrary scalars. | ||
*<math>\langle \mathbf{0}, x \rangle=\langle x,\mathbf{0}\rangle=0.</math> | *<math>\langle \mathbf{0}, x \rangle=\langle x,\mathbf{0}\rangle=0.</math><ref group="Note"><math>\langle \mathbf{0}, x \rangle = \langle x - x,x \rangle = \langle x,x \rangle - \langle x,x \rangle = 0</math> where the right-hand side of the second equality comes from the first argument linearity. <math>\langle x, \mathbf{0} \rangle = 0</math> is also similarly proved by using the conjugate symmetry and the first argument linearity.</ref> | ||
*<math> \langle x, x \rangle</math> is real and nonnegative. | *<math> \langle x, x \rangle</math> is real and nonnegative.<ref group="Note"><math>\langle x,x \rangle = \overline{\langle x,x \rangle}</math> so it is a real number. For <math>x \neq \mathbf{0}</math>, it is a positive real number by the positive-definiteness. For <math>x = \mathbf{0}</math>, it is zero by the 1st basic property above. So, <math>\langle x,x \rangle</math> is real and nonnegative.</ref> | ||
*<math>\langle x, x \rangle = 0</math> if and only if <math>x=\mathbf{0}.</math> | *<math>\langle x, x \rangle = 0</math> if and only if <math>x=\mathbf{0}.</math><ref group="Note">By the 2nd basic property above and the positive-definiteness.</ref> | ||
*<math>\langle x, ay+bz \rangle= \overline a \langle x, y \rangle + \overline b \langle x, z \rangle | *<math>\langle x, ay+bz \rangle= \overline a \langle x, y \rangle + \overline b \langle x, z \rangle</math>, that is conjugate-linearity (for the 2nd argument).<br>This implies that an inner product is a [[sesquilinear form]]. | ||
*<math>\langle x + y, x + y \rangle = \langle x, x \rangle + 2\operatorname{Re}(\langle x, y \rangle) + \langle y, y \rangle,</math> where <math>\operatorname{Re}</math><br>denotes the [[real part]] of its argument. | *<math>\langle x + y, x + y \rangle = \langle x, x \rangle + 2\operatorname{Re}(\langle x, y \rangle) + \langle y, y \rangle,</math> where <math>\operatorname{Re}</math><br>denotes the [[real part]] of its argument. | ||
Over <math>\R</math>, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a ''positive-definite symmetric [[bilinear form]]''. The [[binomial expansion]] of a square becomes | Over <math>\R</math>, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to [[Bilinear form|bilinearity]]. Hence an inner product on a real vector space is a ''positive-definite symmetric [[bilinear form]]''. The [[binomial expansion]] of a square becomes | ||
<math display="block">\langle x + y, x + y \rangle = \langle x, x \rangle + 2\langle x, y \rangle + \langle y, y \rangle .</math> | <math display="block">\langle x + y, x + y \rangle = \langle x, x \rangle + 2\langle x, y \rangle + \langle y, y \rangle .</math> | ||
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=== Convention variant === | === Convention variant === | ||
Some authors, especially in [[physics]] and [[matrix algebra]], prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. [[Bra–ket | Some authors, especially in [[physics]] and [[matrix algebra]], prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. [[Bra–ket notation]] in [[quantum mechanics]] also uses slightly different notation, i.e. <math> \langle \cdot | \cdot \rangle </math>, where <math> \langle x | y \rangle := \left ( y, x \right ) </math>. | ||
==Examples== | ==Examples== | ||
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\begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} | \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} | ||
\right\rangle | \right\rangle | ||
= x^\ | = x^\operatorname{T} y = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots + x_n y_n, | ||
</math> | </math> | ||
where <math>x^{\operatorname{T}}</math> is the [[transpose]] of <math>x.</math> | where <math>x^{\operatorname{T}}</math> is the [[transpose]] of <math>x.</math> | ||
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===Real and complex parts of inner products=== | ===Real and complex parts of inner products=== | ||
Suppose that <math>\langle \cdot, \cdot \rangle</math> is an inner product on <math>V</math> (so it is antilinear in its second argument). The [[polarization identity]] shows that the [[real part]] of the inner product is | Suppose that <math>\langle \cdot, \cdot \rangle</math> is an inner product on <math>V</math> (so it is [[Antilinear_map|antilinear]] in its second argument). The [[polarization identity]] shows that the [[real part]] of the inner product is | ||
<math display=block>\operatorname{Re} \langle x, y \rangle = \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2\right).</math> | <math display=block>\operatorname{Re} \langle x, y \rangle = \frac{1}{4} \left(\|x + y\|^2 - \|x - y\|^2\right).</math> | ||
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* {{annotated link|Orthogonal complement}} | * {{annotated link|Orthogonal complement}} | ||
* {{annotated link|Orthonormal basis}} | * {{annotated link|Orthonormal basis}} | ||
* | * {{annotated link|Riemannian manifold}} | ||
==Notes== | ==Notes== | ||