In mathematics, F. Riesz's theorem (named after Frigyes Riesz) is a 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 (TVS) 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.
Consequences
Throughout, are TVSs (not necessarily Hausdorff) with 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. 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 linear bijection between two finite-dimensional Hausdorff TVSs is a TVS isomorphism. If and are Hausdorff TVS topologies on the same finite-dimensional vector space then A linear map between Hausdorff TVSs is necessarily continuous. In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous. 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. In particular, the convex hull of a compact set is equal to the convex hull of that set. A locally bounded Hausdorff TVS with the Heine-Borel property is finite-dimensional.
See also
References
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