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.

F. Riesz theorem — A Hausdorff TVS $X$ over the field $\mathbb{𝔽}$ ( $\mathbb{𝔽}$ 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, $X$ is TVS-isomorphic to ${\mathbb{𝔽}}^{\text{dim}X}.$

Consequences

Throughout, $F,X,Y$ are TVSs (not necessarily Hausdorff) with $F$ 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.
• 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.

In particular, the range of $L$ is TVS-isomorphic to $X/L^{-1}(0).$

• 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 closed convex hull of that set.
• A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.

See also

• Riesz's lemma

Bibliography

• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.