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.

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Consequences
Throughout, $$F, X, Y$$ are TVSs (not necessarily Hausdorff) with $$F$$ a finite-dimensional vector space. In particular, the range of $$L$$ is TVS-isomorphic to $$X / L^{-1}(0).$$
 * 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.
 * 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 convex hull of that set.
 * A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.