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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Characterization of precompact maps, Schwartz spaces and nuclear spaces

Author: Dan Randtke
Journal: Trans. Amer. Math. Soc. 165 (1972), 87-101
MSC: Primary 46A05; Secondary 47B10
MathSciNet review: 0305009
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Abstract: A general representation theorem for ``precompact'' seminorms on a locally convex space is proven. Using this representation theorem the author derives a representation theorem for precompact maps from one locally convex space into another, that is analogous to the spectral representation theorem for compact maps from one Hilbert space into another and that is applicable to a very extensive class of locally convex spaces. The author uses his representation theorem to derive new characterizations of Schwartz spaces and proves analogous results for nuclear and strongly nuclear spaces.

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Keywords: Bounded map, precompact map, compact map, quasi-Schwartz map, Schwartz map, quasi-nuclear map, nuclear map, type-s map, precompact seminorm, quasi-p-nuclear seminorm, quasi-nuclear seminorm, strongly nuclear seminorm, locally convex space, Schwartz space, nuclear space, strongly nuclear space, strongly summable sequence, $ \mathcal{M}$-topology, $ {c_0}$-extension property
Article copyright: © Copyright 1972 American Mathematical Society