View source for F. Riesz's theorem

{{mergeto|Riesz's lemma|date=July 2024}}
In mathematics, '''F. Riesz's theorem''' (named after [[Frigyes Riesz]]) is an important theorem in [[functional analysis]] that states that a [[Hausdorff space|Hausdorff]] [[topological vector space]] (TVS) is finite-dimensional if and only if it is [[locally compact]]. 
The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned. 

== Statement ==

Recall that a [[topological vector space]] (TVS) <math>X</math> is [[Hausdorff space|Hausdorff]] if and only if the singleton set <math>\{ 0 \}</math> consisting entirely of the origin is a closed subset of <math>X.</math> 
A map between two TVSs is called a '''TVS-isomorphism''' or an '''isomorphism in the category of TVSs''' if it is a [[Linear map|linear]] [[homeomorphism]].

{{Math theorem|name=F. Riesz theorem{{sfn|Narici|Beckenstein|2011|pp=101-105}}{{sfn|Rudin|1991|pp=7-18}}|math_statement=
A [[Hausdorff space|Hausdorff]] TVS <math>X</math> over the field <math>\mathbb{F}</math> ( <math>\mathbb{F}</math> is either the real or complex numbers) is finite-dimensional if and only if it is [[locally compact]] (or equivalently, if and only if there exists a compact neighborhood of the origin). In this case, <math>X</math> is TVS-isomorphic to <math>\mathbb{F}^{\text{dim} X}.</math>
}}

== Consequences ==

Throughout, <math>F, X, Y</math> are TVSs (not necessarily Hausdorff) with <math>F</math> a finite-dimensional vector space.
* Every finite-dimensional vector subspace of a Hausdorff TVS is a closed subspace.{{sfn|Narici|Beckenstein|2011|pp=101-105}}
* All finite-dimensional Hausdorff TVSs are [[Banach space]]s and all norms on such a space are equivalent.{{sfn|Narici|Beckenstein|2011|pp=101-105}}
* '''Closed + finite-dimensional is closed''': If <math>M</math> is a closed vector subspace of a TVS <math>Y</math> and if <math>F</math> is a finite-dimensional vector subspace of <math>Y</math> (<math>Y, M,</math> and <math>F</math> are not necessarily Hausdorff) then <math>M + F</math> is a closed vector subspace of <math>Y.</math>{{sfn|Narici|Beckenstein|2011|pp=101-105}}
* Every vector space isomorphism (i.e. a [[Linear map|linear]] [[bijection]]) between two finite-dimensional Hausdorff TVSs is a [[TVS isomorphism]].{{sfn|Narici|Beckenstein|2011|pp=101-105}}
* '''Uniqueness of topology''': If <math>X</math> is a finite-dimensional vector space and if <math>\tau_1</math> and <math>\tau_2</math> are two Hausdorff TVS topologies on <math>X</math> then <math>\tau_1 = \tau_2.</math>{{sfn|Narici|Beckenstein|2011|pp=101-105}}
* '''Finite-dimensional domain''': A [[linear map]] <math>L : F \to Y</math> between Hausdorff TVSs is necessarily continuous.{{sfn|Narici|Beckenstein|2011|pp=101-105}}
** In particular, every [[linear functional]] of a finite-dimensional Hausdorff TVS is continuous.
* '''Finite-dimensional range''': Any continuous [[surjective]] [[linear map]] <math>L : X \to Y</math> with a Hausdorff finite-dimensional range is an [[open map]]{{sfn|Narici|Beckenstein|2011|pp=101-105}} and thus a [[topological homomorphism]]. 
In particular, the range of <math>L</math> is TVS-isomorphic to <math>X / L^{-1}(0).</math>
* A TVS <math>X</math> (not necessarily Hausdorff) is locally compact if and only if <math>X / \overline{\{ 0 \}}</math> is finite dimensional.
* The [[convex hull]] of a [[compact subset]] of a finite-dimensional Hausdorff TVS is compact.{{sfn|Narici|Beckenstein|2011|pp=101-105}}
** This implies, in particular, that the convex hull of a compact set is equal to the {{em|closed}} convex hull of that set.
* A Hausdorff [[locally bounded]] TVS with the [[Heine-Borel property]] is necessarily finite-dimensional.{{sfn|Rudin|1991|pp=7-18}}

== See also ==

* {{annotated link|Riesz's lemma}}

== References ==

{{reflist|group=note}}
{{reflist}}

==Bibliography==

* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} --> 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

{{Topological vector spaces}}
{{Functional analysis}}

[[Category:Theorems in functional analysis]]
[[Category:Lemmas]]
[[Category:Topological vector spaces]]