On the sequence space and

Authors:
S. A. Schonefeld and W. J. Stiles

Journal:
Trans. Amer. Math. Soc. **225** (1977), 243-257

MSC:
Primary 46A45; Secondary 46A15

DOI:
https://doi.org/10.1090/S0002-9947-1977-0448027-X

MathSciNet review:
0448027

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Abstract: Let and be sequences in the interval , let be the set of all real sequences such that , and let be the set of all real sequences such that where the sup is taken over all permutations of the positive integers. The purpose of this paper is to investigate some of the properties of these spaces. Our results are primarily concerned with (1) conditions which are necessary and/or sufficient for (resp., ) to equal (resp., ), and (2) isomorphic and topological properties of the subspaces of these spaces.

In connection with (1), we show that the following four conditions are equivalent for any sequence which decreases to zero and has . (a) There exists a number such that the series converges; (b) the elements of the sequence satisfy the condition ; (c) the sequence is bounded; and (d) equals . In connection with (2), we show that the following are true when increases to one. (a) contains an infinite-dimensional closed subspace where the -topology and the -topology agree; (b) and contain closed subspaces isomorphic to ; and (c) contains no infinite-dimensional subspace where the -topology agrees with the -topology if and only if

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0448027-X

Keywords:
Sequence spaces,
nonlocally convex spaces,
symmetric spaces,
Schauder bases

Article copyright:
© Copyright 1977
American Mathematical Society