Publication: Geometry of Banach spaces with property β
Loading...
Full text at PDC
Publication Date
1999-12
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Hebrew University Magnes Press
Citations
Exportar
Abstract
We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L 1[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact space
Description
The authors wish to thank S. Troyanski for valuable discussions. Also, they are very grateful to the referee for many helpful suggestions and, in particular, for observing the validity of Proposition 3.2.
UCM subjects
Unesco subjects
Keywords
Citation
M. D. Acosta, F. J. Aguirre and R. Payá,A new sufficient condition for the denseness of norm attaining operators, The Rocky Mountain Journal of Mathematics26 (1996), 407–418.
R. Deville, V. Fonf and P. Hájek,Analytic and polyhedral approximations of convex bodies in separable polyhedral Banach spaces, Israel Journal of Mathematics105 (1998), 139–154.
R. Durier and P. L. Papini,Polyhedral norms and related properties in infinite-dimensional Banach spaces: a survey, Atti del Seminario Matematico Fisico dell’ Università di Modena40 (1992), 623–645.
C. Finet and W. Schachermayer,Equivalent norms in separable Asplund spaces, Studia Mathematica92 (1989), 275–283.
V. Fonf,On polyhedral Banach spaces, Mathematical Notes of the Academy of Sciences of the USSR45 (1989), 809–813. CrossRef
V. Fonf,Three characterizations of polyhedral Banach spaces, Ukrainian Mathematical Journal42 No. 9 (1990), 1145–1148.
C. Franchetti,Lipschitz maps and the geometry of the unit ball in normed spaces, Archiv der Mathematik46(1986), 76–84.
C. Francheti and R. Payá,Banach spaces with strongly subdifferentiable norms, Bolletino. Unione Matematica Italiana7-B (1993), 45–70.
G. Godefroy,Boundaries of a convex set and interpolation sets, Mathematische Annalen277 (1987), 173–194.
G. Godefroy, V. Montesinos and V. Zizler,Strong subdifferentiability of norms and geometry of Banach spaces, Commentationes Mathematicae Universitatis Carolinae36 (1995), 493–502.
M. I. Kadec and V. P. Fonf,Two theorems on the massiveness of boundaries in reflexive Banach spaces, Functional Analysis and its Applications17 (1983), 227–228 (translated from Russian).
V. Klee,Polyhedral sections of convex bodies, Acta Mathematica103 (1960). 243–267.
J. Lindenstrauss,Norm attaining operators, Israel Journal of Mathematics1 (1963), 139–148.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Springer-Verlag, Berlin, 1977.
J. R. Partington,Norm attaining operators, Israel Journal of Mathematics43 (1983), 273–276.
W. Schachermayer,Norm attaining operators and renormings of Banach spaces, Israel Journal of Mathematics44 (1983), 201–212