Basic sequences in non-Schwartz Fréchet spaces

Author:
Steven F. Bellenot

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

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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 -Köthe sequence space, then *E* has the Banach space as a quotient and *E* has a subspace isomorphic to a non-Schwartz -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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0554329-9

Keywords:
Obliquely normalized,
normalized,
unconditional,
conditional and symmetric basic sequences,
Fréchet,
Schwartz,
Montel,
Köthe sequence,
quotient spaces

Article copyright:
© Copyright 1980
American Mathematical Society