F. Riesz's theorem

It has been suggested that this article be merged into Riesz's lemma. (Discuss) Proposed since July 2024.

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.

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 ${\displaystyle \mathbb {F} ^{{\text{dim}}X}.}$

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]

• Riesz's lemma

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