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 | References | Similar Articles | Additional Information

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

**[1]**Zvi Altshuler, P. G. Casazza, and Bor Luh Lin,*On symmetric basic sequences in Lorentz sequence spaces*, Israel J. Math.**15**(1973), 140–155. MR**0328553**, https://doi.org/10.1007/BF02764600**[2]**B. A. Barnes and A. K. Roy,*Boundedness in certain topological linear spaces*, Studia Math.**33**(1969), 147–156. MR**0247402**, https://doi.org/10.4064/sm-33-2-147-156**[3]**C. Bessaga,*A note on universal Banach spaces of a finite dimension*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.**6**(1958), 97–101. MR**0114108****[4]**C. Bessaga and A. Pełczyński,*On bases and unconditional convergence of series in Banach spaces*, Studia Math.**17**(1958), 151–164. MR**0115069**, https://doi.org/10.4064/sm-17-2-151-164**[5]**D. J. H. Garling,*On symmetric sequence spaces*, Proc. London Math. Soc. (3)**16**(1966), 85–106. MR**0192311**, https://doi.org/10.1112/plms/s3-16.1.85**[6]**Joram Lindenstrauss and Lior Tzafriri,*Classical Banach spaces*, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR**0415253****[7]**S. Rolewicz,*On the characterization of Schwartz spaces by properties of the norm*, Studia Math.**20**(1961), 87–92. MR**0137986**, https://doi.org/10.4064/sm-20-1-87-92**[8]**S. Schonefeld and W. Stiles,*On some linear topologies on*(submitted).**[9]**-,*On the inductive limit of*, Studia Math. (to appear).**[10]**S. Simons,*The sequence spaces 𝑙(𝑝_{𝜈}) and 𝑚(𝑝_{𝜈})*, Proc. London Math. Soc. (3)**15**(1965), 422–436. MR**0176325**, https://doi.org/10.1112/plms/s3-15.1.422**[11]**W. J. Stiles,*On non locally 𝑝-convex spaces*, Colloq. Math.**23**(1971), 261–262. MR**0310585**, https://doi.org/10.4064/cm-23-2-261-262

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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