F. Riesz's theorem

Recall that a topological vector space (TVS) $X$ is Hausdorff if and only if the singleton set $\backslash \{\; 0\; \backslash \}$ consisting entirely of the origin is a closed subset of $X.$

A map between two TVSs is called a TVS-isomorphism or an isomorphism in the category of TVSs if it is a 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 TVS $X$ over the field $\backslash mathbb\{F\}$ ( $\backslash mathbb\{F\}$ 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, $X$ is TVS-isomorphic to $\backslash mathbb\{F\}^\{\backslash text\{dim\}\; X\}.$

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Throughout, $F,\; X,\; Y$ are TVSs (not necessarily Hausdorff) with $F$ 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 spaces and all norms on such a space are equivalent.{{sfn|Narici|Beckenstein|2011|pp=101-105}}
• Closed + finite-dimensional is closed: If $M$ is a closed vector subspace of a TVS $Y$ and if $F$ is a finite-dimensional vector subspace of $Y$ ($Y,\; M,$ and $F$ are not necessarily Hausdorff) then $M\; +\; F$ is a closed vector subspace of $Y.${{sfn|Narici|Beckenstein|2011|pp=101-105}}
• Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.{{sfn|Narici|Beckenstein|2011|pp=101-105}}
• Uniqueness of topology: If $X$ is a finite-dimensional vector space and if $\backslash tau\_1$ and $\backslash tau\_2$ are two Hausdorff TVS topologies on $X$ then $\backslash tau\_1\; =\; \backslash tau\_2.${{sfn|Narici|Beckenstein|2011|pp=101-105}}
• Finite-dimensional domain: A linear map $L\; :\; F\; \backslash to\; Y$ 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 $L\; :\; X\; \backslash to\; Y$ 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 $L$ is TVS-isomorphic to $X\; /\; L^\{-1\}(0).$

• A TVS $X$ (not necessarily Hausdorff) is locally compact if and only if $X\; /\; \backslash overline\{\backslash \{\; 0\; \backslash \}\}$ 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}}

• {{annotated link|Riesz's lemma}}

{{reflist|group=note}}

{{reflist}}

• {{Rudin Walter Functional Analysis|edition=2}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}

{{Topological vector spaces}}

{{Functional analysis}}

Category:Theorems in functional analysis

Category:Lemmas

Category:Topological vector spaces