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Transactions of the American Mathematical Society

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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
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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Keywords: Obliquely normalized, normalized, unconditional, conditional and symmetric basic sequences, Fréchet, Schwartz, Montel, Köthe sequence, quotient spaces
Article copyright: © Copyright 1980 American Mathematical Society