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Coverings and renormings
by
Aníbal Moltó
University of Valencia, Spain
Coauthors: J. Orihuela (University of Murcia, Spain), S. Troyanski (University of Sofia, Bulgaria), M. Valdivia (University of Valencia, Spain)
Theorem (Orihuela, Troyanski, Moltó) A Banach space X has an equivalent locally uniformly rotund (LUR) norm iff for every r > 0 there is a countable covering Xn, n=1, 2, ... of X such that for every n and any x in Xn there exists a open half-space H of X containing x in such a way that the diameter of the intersection of H with Xn is less than r.
An easy consequence of this is
Corollary Let T be a linear bounded operator from the Banach space X into a LUR Banach space Y such that for every r > 0 there is a countable covering Xn, n=1, 2, ... of X and a sequence sn > 0 in such a way that the distance from x to y is less than r whenever the distance from Tx to Ty is less than sn, and x, y are points of Xn. Then X is LUR renormable.
The only role of the linearity of T is to guarantee that it transfers slices from Y into X. Nevertheless since we can split X into countable pieces we do not need ``global linearity'' but some sort of ``sigma-linearity''. Conditions on T to have such sort of ``linearity'' are obtained and characterized in terms of r-subdifferentials.
Date received: June 5, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacl-51.