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===Definition as linear functionals=== ...ath>. That is, every element <math>\alpha\in T^*_x\mathcal M</math> is a [[linear map]] ...

...nitial topologies]], often on [[topological vector space]]s or spaces of [[linear operator]]s, for instance on a [[Hilbert space]]. The term is most commonly {{Main|Topologies on spaces of linear maps}} ...

...heorem concerning the dual of a Hilbert space|the theorems relating linear functionals to measures|Riesz–Markov–Kakutani representation theorem|other theorems|Rie === Linear and antilinear maps === ...

...ibration of a circular membrane]]. These modes are [[eigenfunction]]s of a linear operator on a function space, a common construction in functional analysis. ...[Topological space#Definitions|topology]]) and the [[linear transformation|linear function]]s defined on these spaces and suitably respecting these structure ...

...is defined.<ref name="text">{{cite book|first=Frank M.|last=Callier|title=Linear System Theory|location=New York |publisher=Springer-Verlag|year=1991|isbn=0 ...kes <math>\|\,\cdot\,\|</math> continuous and which is compatible with the linear structure of <math>V</math> in the following sense: ...

{{Short description|In mathematics, vector space of linear forms}} ...l vector space''' (or just '''dual space''' for short) consisting of all [[linear form]]s on ''<math>V,</math>'' together with the vector space structure of ...

{{short description|Theorem on extension of bounded linear functionals}} ...there are sufficient [[Continuous function (topology)|continuous]] linear functionals defined on every [[normed vector space]] in order to study the [[dual space ...

...is a [[vector space]] that is a [[subset]] of some larger vector space. A linear subspace is usually simply called a ''subspace'' when the context serves to ...a [[vector space]] over ''K'' for the operations of ''V''. Equivalently, a linear subspace of ''V'' is a [[Empty set|nonempty]] subset ''W'' such that, whene ...

...ac notation''' is a [[mathematical notation]] for [[linear algebra]] and [[linear operator]]s on [[complex vector space]]s together with their [[dual space|d .../math> to a number in the complex plane <math>\Complex</math>. Letting the linear functional <math> \langle f|</math> act on a vector <math>|v\rangle</math> ...

{{short description|Mathematical function, in linear algebra}} {{redirect|Linear transformation|fractional linear transformations|Möbius transformation}} ...

...unctions act by integration against a test function, but many other linear functionals do not arise in this way, and these are the "generalized functions". There Distributions are a class of [[linear functional]]s that map a set of ''test functions'' (conventional and [[well ...

...e same value. In simple problems, this method is unlikely to be used, as [[linear]] interpolation (see below) is almost as easy, but in higher-dimensional [[ ===Linear interpolation=== ...

...le= Convex Functional Analysis | date= 2005 | chapter=Lower Semicontinuous Functionals | publisher=Birkhäuser-Verlag | series=Systems & Control: Foundations & App ...f>{{Cite book |last=Giaquinta |first=Mariano |title=Mathematical analysis: linear and metric structures and continuity |date=2007 |publisher=Birkhäuser |othe ...

In [[mathematics]], a '''topological vector space''' (also called a '''linear topological space''' and commonly abbreviated '''TVS''' or '''t.v.s.''') is ...spaces are spaces of [[Function (mathematics)|function]]s, or [[Linear map|linear operators]] acting on topological vector spaces, and the topology is often ...

...me="Strang5">{{cite book|last=Strang|first=Gilbert|date=19 July 2005|title=Linear Algebra and its Applications|edition=4th|chapter=3.2|publisher=Cengage Lear ...tional.}}</ref><ref>{{cite book|last1=Axler|first1=Sheldon|date=2015|title=Linear Algebra Done Right, 3rd Ed.|publisher=Springer International Publishing|isb ...

In [[linear algebra]], the '''trace''' of a [[square matrix]] {{math|'''A'''}}, denoted ...s]] have the same trace. As a consequence, one can define the trace of a [[linear operator]] mapping a finite-dimensional [[vector space]] into itself, since ...

...bly many copies of <math>\Reals</math> (this homeomorphism need not be a [[linear map]]).<ref>{{harvnb|Bessaga|Pełczyński|1975|p=189}}</ref>{{sfn|Anderson|Sc ===Linear operators, isomorphisms=== ...

..._0</math> norm was established by [[Stefan Banach|Banach]]'s ''[[Theory of Linear Operations]]''. The [[F-space|space]] of sequences has a complete metric to ...cite book|last1=Garling|first1=D. J. H.|title=Inequalities: A Journey into Linear Analysis|date=2007|publisher=Cambridge University Press|isbn=978-0-521-8762 ...

{{Short description|Algebraic structure in linear algebra}} {{redirect|Linear space |a structure in incidence geometry|Linear space (geometry)}} ...

...ained from the density operator by a choice of an [[orthonormal]] [[basis (linear algebra)|basis]] in the underlying space.<ref>{{cite book | last=Ballentine ...is a [[projection (linear algebra)|projection]], in particular of [[Rank (linear algebra)|rank]] one. ...