F. Riesz's theorem

Template:Mergeto In mathematics, 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.

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.Template:Sfn
• All finite-dimensional Hausdorff TVSs are Banach spaces and all norms on such a space are equivalent.Template:Sfn
• 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 $Y.$Template:Sfn
• Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.Template:Sfn
• 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 ${\tau }_1={\tau }_2.$Template:Sfn
• Finite-dimensional domain: A linear map $L:F\to Y$ between Hausdorff TVSs is necessarily continuous.Template:Sfn
  • 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 mapTemplate:Sfn 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.Template:Sfn
  • This implies, in particular, that the convex hull of a compact set is equal to the Template:Em convex hull of that set.
• A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.Template:Sfn

See also

• Template:Annotated link

Bibliography

• Template:Rudin Walter Functional Analysis
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels