DELETING DIFFEOMORPHISMS AND STARLIKE BODIES IN BANACH SPACES

L. ALEJANDRO MONTESINOS MATILLA

Departamento de Análisis Matemático. Facultad de Matemáticas. Universidad Complutense de Madrid. España

Abstract

This talk aims to show the theory of topological negligibility in its historical context, and to explore, with some detail, both the way in which it has developed and its interesting applications in several branches of mathematics. Given a topological space X and a subset K of X, we say that K is topologically negligible if there exists a homeomorphism between X and X \ K. This homeomorphism, which we call deleting, is usually required to be the identity outside a given neighborhood of K. In 1953, V. Klee proved that if X is a non-reflexive Banach space or an infinite-dimensional L^{p} space, every K compact subset of X is topologically negligible. Later on, several mathematicians (C. Bessaga, T. Dobrowolski and others) studied the smooth negligibility, where, by discarding deleting homeomorphims, the attention is paid to the notion of deleting diffeomorphims. For the time being, the deepest results about smooth negligiblity are due to D. Azagra and T. Doborwolski, who showed that for every Banach space (X, ∥ · ∥) with a C^{p}-smooth norm ρ, compacta of X are negligible via a C^{p}-diffeomorphism. Now, in a joint work with D. Azagra, we have proved, by exploiting starlike bodies, some other results that contribute to understand in what Banach spaces, compacta and points are smooth negligible. For instance, we have proved that in every Banach space X with smooth partitions of unity, weak compacta of X are smooth negligible. In addition, it has been established that if X is an infinite-dimensional separable Banach space with a Schauder basis, and X has a C^{p}- smooth bump function, then for every compact subset K and every open subset U of X with K ⊂ U, there exists a C^{p}-diffeomorphism h : X → X \ K such that h is the identity on X \ U.

Finally, it is important to remark that this kind of results about topological and smooth negligibility has found interesting applications in several branches of mathematics, which include fixed point theory, strange fenomena concerning EDO’s, and approximate strong versions of the Morse-Sard theorem.