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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 \ 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. 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 clo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematician who made fundamental contributions to functional analysis, as did his younger brother Marcel Riesz. Life and career He was born into a Jewish family in Győr, Austria-Hungary and died in Budapest, Hungary. Between 1911 and 1919 he was a professor at the Franz Joseph University in Kolozsvár, Austria-Hungary. The post-WW1 Treaty of Trianon transferred former Austro-Hungarian territory including Kolozsvár to the Kingdom of Romania, whereupon Kolozsvár's name changed to Cluj and the University of Kolozsvár moved to Szeged, Hungary, becoming the University of Szeged. Then, Riesz was the rector and a professor at the University of Szeged, as well as a member of the Hungarian Academy of Sciences. and the Polish Academy of Lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |