Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-21T05:24:13.885Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally uniformly rotund norms and Markuschevich bases

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

A. C. Yorke
A. C. Yorke
Affiliation:
Department of MathematicsUniversity of New EnglandArmidale, N.S.W. 2351, Australia
Rights & Permissions [Opens in a new window]

Abstract

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.

If a Banach space E admits a Markuschevich basis, then E can be renormed to be locally uniformly rotund. When the coefficient space of the basis is 1-norming, and this norm is very smooth, E is weakly compactly generated.

MSC classification

Secondary: 46B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 3 , May 1979 , pp. 371 - 377
Copyright
Copyright © Australian Mathematical Society 1979

References

Amir, D. and Lindenstrauss, J. (1968), ‘The structure of weakly compact sets in Banach spaces’, Ann. of Math. (2) 88, 35–46.CrossRefGoogle Scholar
Asplund, E. (1967), ‘Averaged norms’, Israel J. Math. 5, 227–233.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. (1958), Linear operators, Part I (Interscience, New York).Google Scholar
Dyer, J. (1968), ‘Generalized Markuschevich bases’, Israel J. Math. 7, 51–59.CrossRefGoogle Scholar
Giles, J. R. (1971), ‘On a characterization of the differentiability of the norm of a normed linear space’, J. Austral. Math. Soc. Ser. A 12, 106–114.CrossRefGoogle Scholar
Giles, J. R. (1975), ‘On smoothness of the Banach space embedding’, Bull. Austral. Math. Soc. 13, 69–74.CrossRefGoogle Scholar
Johnson, W. B. (1970a), ‘Markuschevich bases and duality theory’, Trans. Amer. Math. Soc. 149, 171–177.CrossRefGoogle Scholar
Johnson, W. B. (1970b), ‘No infinite dimensional P space admits a Markuschevich basis’, Proc. Amer. Math. Soc. 28, 467–468.Google Scholar
Klee, V. (1953), ‘Convex bodies and periodic homeomorphisms in Hilbert spaces’, Trans. Amer. Math. Soc. 74, 10–43.CrossRefGoogle Scholar
Lindenstrauss, J. (1972), ‘Weakly compact sets—their topological properties and the Banach spaces they generate’, Ann. of Math. Studies 69, 235–273.Google Scholar
Lovaglia, A. (1955), ‘Locally uniformly convex Banach spaces’, Trans. Amer. Math. Soc. 78, 225–238.CrossRefGoogle Scholar
Tacon, D. C. (1970), ‘The conjugate of a smooth Banach space’, Bull. Austral. Math. Soc. 2, 415–425.CrossRefGoogle Scholar
Troyanski, S. (1971), ‘On locally uniformly convex and differentiable norms in certain non-spearable Banach spaces’, Studia Math. 37, 173–180.CrossRefGoogle Scholar
Vašak, L. (1973), ‘Gerneralized Markuschevich bases’, Comm. Math. Univ. Carol. 14, 92, 215–222.Google Scholar
Zizler, V. and John, K. (1974a), ‘Smoothness and its equivalents in weakly compactly generated spaces’, J. Functional Analysis 15, 1–11.Google Scholar
Zizler, V. and John, K. (1974b), ‘Some remarks on non-separable Banach spaces with Markuschevich bases’, Comm. Math. Univ. Carol. 15 (4), 679–691.Google Scholar