Infunctional analysis and related areas of mathematics, Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets. These spaces were introduced by Alexander Grothendieck.
AHausdorff locally convex space X with continuous dual , X is called a Schwartz space if it satisfies any of the following equivalent conditions:[1]
Every quasi-complete Schwartz space is a semi-Montel space. Every Fréchet Schwartz space is a Montel space.[2]
The strong dual space of a complete Schwartz space is an ultrabornological space.
Every infinite-dimensional normed spaceisnot a Schwartz space.[2]
There exist Fréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces.[2]
Boundedness and bornology
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Operators |
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Subsets |
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Related spaces |
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Topological vector spaces (TVSs)
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Main results |
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Maps |
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Types of sets |
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Set operations |
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Types of TVSs |
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