F. Riesz's theorem
# F. Riesz's theorem

F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in functional analysis that states that a 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) $X$ is Hausdorff if and only if the singleton set $\{0\}$ 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.

F. Riesz theorem — A Hausdorff TVS $X$ over the field $\mathbb {F}$ ( $\mathbb {F}$ 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 $\mathbb {F} ^{{\text{dim}}X}.$

## Consequences

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.
* All finite-dimensional Hausdorff TVSs are Banach spaces and all norms on such a space are equivalent.
* 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 $Y.$
* Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.
* Uniqueness of topology: If $X$ is a finite-dimensional vector space and if $\tau _{1}$ and $\tau _{2}$ are two Hausdorff TVS topologies on $X$ then $\tau _{1}=\tau _{2}.$
* Finite-dimensional domain: A linear map $L:F\to Y$ between Hausdorff TVSs is necessarily continuous.
   * In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous.
* Finite-dimensional range: Any continuous surjective linear map $L:X\to Y$ with a Hausdorff finite-dimensional range is an open map 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 $X/{\overline {\{0\}}}$ is finite dimensional.
* The convex hull of a compact subset of a finite-dimensional Hausdorff TVS is compact.
   * This implies, in particular, that the convex hull of a compact set is equal to the closed convex hull of that set.
* A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.
