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 𝔽 ( 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 𝔽^{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 𝜏₁ and 𝜏₂ are two Hausdorff TVS topologies on X then 𝜏₁ = 𝜏₂.
• Finite-dimensional domain: A linear map L : F → 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 → 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

Sources

• 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.