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) is Hausdorff if and only if the singleton set consisting entirely of the origin is a closed subset of 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 over the field ( 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, is TVS-isomorphic to

Consequences

Throughout, are TVSs (not necessarily Hausdorff) with 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 is a closed vector subspace of a TVS and if is a finite-dimensional vector subspace of ( and are not necessarily Hausdorff) then is a closed vector subspace of [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 is a finite-dimensional vector space and if and are two Hausdorff TVS topologies on then [1]
  • Finite-dimensional domain: A linear map 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 with a Hausdorff finite-dimensional range is an open map[1] and thus a topological homomorphism.

In particular, the range of is TVS-isomorphic to

  • A TVS (not necessarily Hausdorff) is locally compact if and only if 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

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.
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