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) X {\displaystyle X} is Hausdorff if and only if the singleton set { 0 } {\displaystyle \{0\}} consisting entirely of the origin is a closed subset of X . {\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.
Consequences Throughout, F , X , Y {\displaystyle F,X,Y} are TVSs (not necessarily Hausdorff) with F {\displaystyle F} 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 M {\displaystyle M} is a closed vector subspace of a TVS Y {\displaystyle Y} and if F {\displaystyle F} is a finite-dimensional vector subspace of Y {\displaystyle Y} then M + F {\displaystyle M+F} is a closed vector subspace of Y . {\displaystyle Y.}
Every linear bijection between two finite-dimensional Hausdorff TVSs is a TVS isomorphism. If τ 1 {\displaystyle \tau _{1}} and τ 2 {\displaystyle \tau _{2}} are Hausdorff TVS topologies on the same finite-dimensional vector space then τ 1 = τ 2 . {\displaystyle \tau _{1}=\tau _{2}.}
A linear map L : F → Y {\displaystyle L:F\to Y} between Hausdorff TVSs is necessarily continuous. In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous. Any continuous surjective linear map L : X → Y {\displaystyle L:X\to Y} with a Hausdorff finite-dimensional range is an open map and thus a topological homomorphism. In particular, the range of L {\displaystyle L} is TVS-isomorphic to X / L − 1 ( 0 ) . {\displaystyle X/L^{-1}(0).}
A TVS X {\displaystyle X} is locally compact if and only if X / { 0 } ¯ {\displaystyle X/{\overline {\{0\}}}} 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 closed convex hull of that set. A locally bounded Hausdorff TVS with the Heine-Borel property is finite-dimensional.
See also Riesz's lemma – Mathematics lemma in functional analysis
References
Bibliography Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.