Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-02-02T22:50:29.866Z Has data issue: false hasContentIssue false
  • English
  • Français

On Nonreflexive Banach Spaces Which Contain No c0 or lp

Published online by Cambridge University Press:  20 November 2018

P. G. Casazza ,
Bor-Luh Lin  and
R. H. Lohman
P. G. Casazza
Affiliation:
University of Alabama, Huntsville, Alabama
Bor-Luh Lin
Affiliation:
University of Iowa, Iowa City, Iowa
R. H. Lohman
Affiliation:
Kent State University, Kent, Ohio
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The first infinite-dimensional reflexive Banach space X such that no subspace of X is isomorphic to c0 or lp, 1 ≦ p < ∞, was constructed by Tsirelson [8]. In fact, he showed that there exists a Banach space with an unconditional basis which contains no subsymmetric basic sequence and which contains no superreflexive subspace. Subsequently, Figiel and Johnson [4] gave an analytical description of the conjugate space T of Tsirelson's example and showed that there exists a reflexive Banach space with a symmetric basis which contains no superreflexive subspace; a uniformly convex space with a symmetric basis which contains no isomorphic copy of lp, 1 < p < ∞; and a uniformly convex space which contains no subsymmetric basic sequence and hence contains no isomorphic copy of lp, 1 < p < ∞. Recently, Altshuler [2] showed that there is a reflexive Banach space with a symmetric basis which has a unique symmetric basic sequence up to equivalence and which contains no isomorphic copy of lp, 1 < p < ∞.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1382 - 1389
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Altshuler, Z., Characterization of C0 and lp among Banach spaces with a symmetric basis, Israel J. Math. 24 (1976), 39–44.Google Scholar
2. Altshuler, Z., A Banach space with a symmetric basis which contains no lp or c0 and all its symmetric basis sequences are equivalent, Compositio Math. 35 (1977), 189–195.Google Scholar
3. Casazza, P. G. and Lohman, R. H., A general construction of spaces of the type of R. C. James, Can. J. Math. 27 (1975), 1263–1270.Google Scholar
4. Figiel, T. and Johnson, W. B., A uniformly convex Banach space which contains no lp, Compsotio Math. 29 (1974), 179–190.Google Scholar
5. Herman, R. and Whitley, R., An example concerning reflexivity, Studia Math. 28 (1967), 289–294.Google Scholar
6. James, R. C., Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550.Google Scholar
7. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, I (Springer-Verlag, 1977).CrossRefGoogle Scholar
8. Tsirelson, B. S., Not every Banach space contains lp or c0, Functional Anal. Appl. 8 (1974), 138–141.Google Scholar