F. Riesz's theorem

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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

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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

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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

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References

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  1. ^ a b c d e f g h i Narici & Beckenstein 2011, pp. 101–105.
  2. ^ a b Rudin 1991, pp. 7–18.

Bibliography

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  • 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.