Renormings and coverings in Banach spaces
by
J. Orihuela
Murcia University
Coauthors: A. Moltó (University of Valencia), S. Troyanski (University of Sofia), M. Valdivia (University of Valencia)

The existence of locally uniformly rotund norms on a given non separable Banach space has a considerable impact in his geometry and topology. For instance every discrete family of sets for the norm topology {A_i; i in I} can be decomposed with a countable partition:

Ai = È {Ai, n; n = 1, 2, ...}

in such a way that every one of the families {A_{i, n}; n = 1, 2, ...} for i in I verifies the following ``strong discreteness property":

Ai, n Ç conv   {Aj, n, j =/= i, j in I} = \emptyset

that we call "half-space isolated family" because of the Hahn Banach theorem. Consequently every Banach space with a locally uniformly rotund norm has a network for the norm topology which is \sigma -half-space isolated. This condition indeed characterizes the Banach spaces admitting an equivalent locally uniformly rotund norm [1], and it has been the beginning of our "topological approach" to renormings. We shall illustrate how the former characterization has been leading to a series of positive results for locally uniformly rotund renormability of a given Banach space. For instance we will show how a weakly locally uniformly rotund norm gives us a \sigma -half-space isolated network for the norm topology and consequently that it is locally uniformly rotund renormable, [2]. The same is true for a Banach space with the Krein Milman property if it has a Kadec norm [3]. Therefore in spaces with the RNP to have an equivalent locally uniformly rotund norm or to have an equivalent Kadec norm are the same. We shall connect our results with midpoint locally uniformly rotund renormings [4], decomposition and transfer methods, [5].

References

[1] A. Moltó, J. Orihuela, and S. Troyanski, Locally uniformly rotund renorming and fragmentability. Proc. London Math. Soc., 75, (1997), 619-640.

[2] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia, On weakly locally uniformly rotund Banach spaces. J. Functional Anal. 163, (1999), 252-271

[3] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia, Kadec and Krein-Milman properties. To appear in C.R.A.S. Paris

[4] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia, Midpoint locally uniformly rotundity and a decomposition method for renorming. To appear in Quarterly Journal Oxford.

[5] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia, Non linear transfer technique. Preprint.

(T)

Date received: April 15, 2000