Английская Википедия: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>X</math> is Hausdorff if and only if the singleton set <math>\{ 0 \}</math> consisting entirely of the origin is a closed subset of <math>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.
Consequences
Throughout, <math>F, X, Y</math> are TVSs (not necessarily Hausdorff) with <math>F</math> a finite-dimensional vector space.
- Every finite-dimensional vector subspace of a Hausdorff TVS is a closed subspace.Шаблон:Sfn
- All finite-dimensional Hausdorff TVSs are Banach spaces and all norms on such a space are equivalent.Шаблон:Sfn
- Closed + finite-dimensional is closed: If <math>M</math> is a closed vector subspace of a TVS <math>Y</math> and if <math>F</math> is a finite-dimensional vector subspace of <math>Y</math> (<math>Y, M,</math> and <math>F</math> are not necessarily Hausdorff) then <math>M + F</math> is a closed vector subspace of <math>Y.</math>Шаблон:Sfn
- Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.Шаблон:Sfn
- Uniqueness of topology: If <math>X</math> is a finite-dimensional vector space and if <math>\tau_1</math> and <math>\tau_2</math> are two Hausdorff TVS topologies on <math>X</math> then <math>\tau_1 = \tau_2.</math>Шаблон:Sfn
- Finite-dimensional domain: A linear map <math>L : F \to Y</math> between Hausdorff TVSs is necessarily continuous.Шаблон:Sfn
- In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous.
- Finite-dimensional range: Any continuous surjective linear map <math>L : X \to Y</math> with a Hausdorff finite-dimensional range is an open mapШаблон:Sfn and thus a topological homomorphism.
In particular, the range of <math>L</math> is TVS-isomorphic to <math>X / L^{-1}(0).</math>
- A TVS <math>X</math> (not necessarily Hausdorff) is locally compact if and only if <math>X / \overline{\{ 0 \}}</math> is finite dimensional.
- The convex hull of a compact subset of a finite-dimensional Hausdorff TVS is compact.Шаблон:Sfn
- This implies, in particular, that the convex hull of a compact set is equal to the Шаблон:Em convex hull of that set.
- A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.Шаблон:Sfn
See also
References
Bibliography
- Шаблон:Rudin Walter Functional Analysis
- Шаблон:Narici Beckenstein Topological Vector Spaces
- Шаблон:Schaefer Wolff Topological Vector Spaces
- Шаблон:Trèves François Topological vector spaces, distributions and kernels
Шаблон:Topological vector spaces Шаблон:Functional analysis
