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) [math]\displaystyle{ X }[/math] is Hausdorff if and only if the singleton set [math]\displaystyle{ \{ 0 \} }[/math] consisting entirely of the origin is a closed subset of [math]\displaystyle{ X. }[/math] 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 [math]\displaystyle{ X }[/math] over the field [math]\displaystyle{ \mathbb{F} }[/math] ( [math]\displaystyle{ \mathbb{F} }[/math] 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, [math]\displaystyle{ X }[/math] is TVS-isomorphic to [math]\displaystyle{ \mathbb{F}^{\text{dim} X}. }[/math]

Consequences

Throughout, [math]\displaystyle{ F, X, Y }[/math] are TVSs (not necessarily Hausdorff) with [math]\displaystyle{ F }[/math] 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 [math]\displaystyle{ M }[/math] is a closed vector subspace of a TVS [math]\displaystyle{ Y }[/math] and if [math]\displaystyle{ F }[/math] is a finite-dimensional vector subspace of [math]\displaystyle{ Y }[/math] ([math]\displaystyle{ Y, M, }[/math] and [math]\displaystyle{ F }[/math] are not necessarily Hausdorff) then [math]\displaystyle{ M + F }[/math] is a closed vector subspace of [math]\displaystyle{ Y. }[/math]
• Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.
• Uniqueness of topology: If [math]\displaystyle{ X }[/math] is a finite-dimensional vector space and if [math]\displaystyle{ \tau_1 }[/math] and [math]\displaystyle{ \tau_2 }[/math] are two Hausdorff TVS topologies on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ \tau_1 = \tau_2. }[/math]
• Finite-dimensional domain: A linear map [math]\displaystyle{ L : F \to Y }[/math] 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 [math]\displaystyle{ L : X \to Y }[/math] with a Hausdorff finite-dimensional range is an open map and thus a topological homomorphism.

In particular, the range of [math]\displaystyle{ L }[/math] is TVS-isomorphic to [math]\displaystyle{ X / L^{-1}(0). }[/math]

• A TVS [math]\displaystyle{ X }[/math] (not necessarily Hausdorff) is locally compact if and only if [math]\displaystyle{ X / \overline{\{ 0 \}} }[/math] 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.