Basic sequences in non-Schwartz Fréchet spaces

Bellenot, Steven F.

doi:10.1090/S0002-9947-1980-0554329-9

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 ${l_p}$-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.

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