# 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) $\displaystyle{ X }$ is Hausdorff if and only if the singleton set $\displaystyle{ \{ 0 \} }$ consisting entirely of the origin is a closed subset of $\displaystyle{ X. }$ 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[1][2] — A Hausdorff TVS $\displaystyle{ X }$ over the field $\displaystyle{ \mathbb{F} }$ ( $\displaystyle{ \mathbb{F} }$ 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, $\displaystyle{ X }$ is TVS-isomorphic to $\displaystyle{ \mathbb{F}^{\text{dim} X}. }$

## Consequences

Throughout, $\displaystyle{ F, X, Y }$ are TVSs (not necessarily Hausdorff) with $\displaystyle{ F }$ a finite-dimensional vector space.

• Every finite-dimensional vector subspace of a Hausdorff TVS is a closed subspace.[1]
• All finite-dimensional Hausdorff TVSs are Banach spaces and all norms on such a space are equivalent.[1]
• Closed + finite-dimensional is closed: If $\displaystyle{ M }$ is a closed vector subspace of a TVS $\displaystyle{ Y }$ and if $\displaystyle{ F }$ is a finite-dimensional vector subspace of $\displaystyle{ Y }$ ($\displaystyle{ Y, M, }$ and $\displaystyle{ F }$ are not necessarily Hausdorff) then $\displaystyle{ M + F }$ is a closed vector subspace of $\displaystyle{ Y. }$[1]
• Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.[1]
• Uniqueness of topology: If $\displaystyle{ X }$ is a finite-dimensional vector space and if $\displaystyle{ \tau_1 }$ and $\displaystyle{ \tau_2 }$ are two Hausdorff TVS topologies on $\displaystyle{ X }$ then $\displaystyle{ \tau_1 = \tau_2. }$[1]
• Finite-dimensional domain: A linear map $\displaystyle{ L : F \to Y }$ between Hausdorff TVSs is necessarily continuous.[1]
• In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous.
• Finite-dimensional range: Any continuous surjective linear map $\displaystyle{ L : X \to Y }$ with a Hausdorff finite-dimensional range is an open map[1] and thus a topological homomorphism.

In particular, the range of $\displaystyle{ L }$ is TVS-isomorphic to $\displaystyle{ X / L^{-1}(0). }$

• A TVS $\displaystyle{ X }$ (not necessarily Hausdorff) is locally compact if and only if $\displaystyle{ X / \overline{\{ 0 \}} }$ is finite dimensional.
• The convex hull of a compact subset of a finite-dimensional Hausdorff TVS is compact.[1]
• 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.[2]