×

Kadec and Krein-Milman properties. (English. Abridged French version) Zbl 0983.46012

The norm of a Banach space \(X\) is said to have the Krein-Milman property if every nonempty convex bounded norm-closed subset of its unit sphere \(S_X\) has at least one extreme point. The norm of \(X\) has the Kadec property provided the identity mapping \((S_X ,\)weak\() \to (S_X,\|\|)\) is continuous on \(S_X\). One says also that every point of \(S_X\) is a point of weak continuity.
The main goal of the present paper is to prove that a Banach space whose norm has both the Kadec and Krein-Milman properties admits a locally uniformly rotund (LUR) renorming. S. Troyanski [Math. Ann. 271, 305-313 (1985; Zbl 0557.46014)], proved that a Banach space whose norm is rotund and has the Kadec property admits a LUR renorming. R. Haydon [Proc. Lond. Math. Soc. 78, 549-584 (1999; Zbl 1036.46003)], has shown that the Kadec property of the norm does not imply the existence of an equivalent rotund norm. Recently, M. Raja [Mathematika (2000; to appear)] proved that if the norm in \(X^*\) has the weak\(^*\) Kadec property then \(X^*\) admits a LUR renorming.

MSC:

46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46B03 Isomorphic theory (including renorming) of Banach spaces
PDFBibTeX XMLCite
Full Text: DOI