F. Riesz's theorem

In mathematics, F. Riesz's theorem is an important theorem in functional analysis that states that a Hausdorff topological vector space 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 is Hausdorff if and only if the singleton set consisting entirely of the origin is a closed subset of
A map between two TVSs is called a TVS-isomorphism or an isomorphism in the category of TVSs if it is a linear homeomorphism.

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 is a closed vector subspace of a TVS and if is a finite-dimensional vector subspace of then is a closed vector subspace of
Every vector space isomorphism between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.Uniqueness of topology: If is a finite-dimensional vector space and if and are two Hausdorff TVS topologies on then Finite-dimensional domain: A linear map 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 with a Hausdorff finite-dimensional range is an open map and thus a topological homomorphism.

In particular, the range of is TVS-isomorphic to

A TVS is locally compact if and only if 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 convex hull of that set.
A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.