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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Basic sequences in non-Schwartz Fréchet spaces
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by Steven F. Bellenot PDF
Trans. Amer. Math. Soc. 258 (1980), 199-216 Request permission

Abstract:

Obliquely normalized basic sequences are defined and used to characterize non-Schwartz-Fréchet spaces. It follows that each non-Schwartz-Fréchet space E has a non-Schwartz subspace with a basis and a quotient which is not Montel (which has a normalized basis if E is separable). Stronger results are given when more is known about E, for example, if E is a subspace of a Fréchet ${l_p}$-Köthe sequence space, then E has the Banach space ${l_p}$ as a quotient and E has a subspace isomorphic to a non-Schwartz ${l_p}$-Köthe sequence space. Examples of Fréchet-Montel spaces which are not subspaces of any Fréchet space with an unconditional basis are given. The question of the existence of conditional basic sequences in non-Schwartz-Fréchet spaces is reduced to questions about Banach spaces with symmetric bases. Nonstandard analysis is used in some of the proofs and a new nonstandard characterization of Schwartz spaces is given.
References
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 258 (1980), 199-216
  • MSC: Primary 46A35; Secondary 03H05
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0554329-9
  • MathSciNet review: 554329