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Concept
Montel space
Summary
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space in which every closed and bounded subset is compact.
A topological vector space (TVS) has the if every closed and bounded subset is compact.
A is a barrelled topological vector space with the Heine–Borel property. Equivalently, it is an infrabarrelled semi-Montel space where a Hausdorff locally convex topological vector space is called a or if every bounded subset is relatively compact.
A subset of a TVS is compact if and only if it is complete and totally bounded.
A is a Fréchet space that is also a Montel space.
A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual is strongly convergent.
A Fréchet space is a Montel space if and only if every bounded continuous function sends closed bounded absolutely convex subsets of to relatively compact subsets of
Moreover, if denotes the vector space of all bounded continuous functions on a Fréchet space then is Montel if and only if every sequence in that converges to zero in the compact-open topology also converges uniformly to zero on all closed bounded absolutely convex subsets of
Semi-Montel spaces
A closed vector subspace of a semi-Montel space is again a semi-Montel space. The locally convex direct sum of any family of semi-Montel spaces is again a semi-Montel space. The inverse limit of an inverse system consisting of semi-Montel spaces is again a semi-Montel space. The Cartesian product of any family of semi-Montel spaces (resp. Montel spaces) is again a semi-Montel space (resp. a Montel space).
Montel spaces
The strong dual of a Montel space is Montel.
A barrelled quasi-complete nuclear space is a Montel space.
Every product and locally convex direct sum of a family of Montel spaces is a Montel space.
The strict inductive limit of a sequence of Montel spaces is a Montel space.
Official source
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