Basic Sequences in Non-Schwartz-Frechet Spaces
Copyright policy )Basic Sequences in Non-Schwartz-Frechet Spaces
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 {l_p} -Köthe sequence space, then E has the Banach space l p {l_p} as a quotient and E has a subspace isomorphic to a non-Schwartz l p {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.
Summability and bases in topological vector spaces, Köthe sequence space, Frechet- Montel spaces, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, nonstandard analysis, obliquely normalized, Nonstandard analysis, unconditional basis, nonstandard characterization of Schwartz spaces, quotient spaces, Locally convex Fréchet spaces and (DF)-spaces, Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.), non-Schwartz-Frechet spaces, Sequence spaces (including Köthe sequence spaces), not Montel
Summability and bases in topological vector spaces, Köthe sequence space, Frechet- Montel spaces, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, nonstandard analysis, obliquely normalized, Nonstandard analysis, unconditional basis, nonstandard characterization of Schwartz spaces, quotient spaces, Locally convex Fréchet spaces and (DF)-spaces, Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.), non-Schwartz-Frechet spaces, Sequence spaces (including Köthe sequence spaces), not Montel
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