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) ${\displaystyle X}$ is Hausdorff if and only if the singleton set ${\displaystyle \{0\}}$ consisting entirely of the origin is a closed subset of ${\displaystyle X.}$ A map between two TVSs is called a TVS-isomorphism or an isomorphism in the category of TVSs if it is a linear homeomorphism.

F. Riesz theorem ^{[1] [2]}  — A Hausdorff TVS ${\displaystyle X}$ over the field ${\displaystyle \mathbb {F} }$ ( ${\displaystyle \mathbb {F} }$ 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, ${\displaystyle X}$ is TVS-isomorphic to ${\displaystyle \mathbb {F} ^{{\text{dim}}X}.}$

Consequences

Throughout, ${\displaystyle F,X,Y}$ are TVSs (not necessarily Hausdorff) with ${\displaystyle F}$ a finite-dimensional vector space.

• Every finite-dimensional vector subspace of a Hausdorff TVS is a closed subspace. ^[1]
• All finite-dimensional Hausdorff TVSs are Banach spaces and all norms on such a space are equivalent. ^[1]
• Closed + finite-dimensional is closed: If ${\displaystyle M}$ is a closed vector subspace of a TVS ${\displaystyle Y}$ and if ${\displaystyle F}$ is a finite-dimensional vector subspace of ${\displaystyle Y}$ (${\displaystyle Y,M,}$ and ${\displaystyle F}$ are not necessarily Hausdorff) then ${\displaystyle M+F}$ is a closed vector subspace of ${\displaystyle Y.}$ ^[1]
• Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism. ^[1]
• Uniqueness of topology: If ${\displaystyle X}$ is a finite-dimensional vector space and if ${\displaystyle \tau _{1}}$ and ${\displaystyle \tau _{2}}$ are two Hausdorff TVS topologies on ${\displaystyle X}$ then ${\displaystyle \tau _{1}=\tau _{2}.}$ ^[1]
• Finite-dimensional domain: A linear map ${\displaystyle L:F\to Y}$ between Hausdorff TVSs is necessarily continuous. ^[1]
  • In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous.
• Finite-dimensional range: Any continuous surjective linear map ${\displaystyle L:X\to Y}$ with a Hausdorff finite-dimensional range is an open map ^[1] and thus a topological homomorphism.

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

• A TVS ${\displaystyle X}$ (not necessarily Hausdorff) is locally compact if and only if ${\displaystyle X/{\overline {\{0\}}}}$ is finite dimensional.
• The convex hull of a compact subset of a finite-dimensional Hausdorff TVS is compact. ^[1]
  • 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. ^[2]

See also

• Riesz's lemma

References

1. Narici & Beckenstein 2011, pp. 101–105.
2. Rudin 1991, pp. 7–18.

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.