Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications

Abstract

This article deals with smooth removability of compact sets in infinite-dimensional Banach spaces. The main result states that ifX is an infinite-dimensional Banach space which has a not necessarily equivalent Cp-smooth norm and K is a compact subset of X, then X and X r K are Cp diffeomorphic. The proof relies on the construction of a “deleting path” through a nontrivial refinement of Bessaga’s incomplete-norm technique. However, norms are not at present available and the construction requires the use of asymmetric functionals. The noncompleteness of such functionals relies in turn on James’ theorem on existence of linear functionals which do not attain their norm on every nonreflexive space. Applications are given which show that several important theorems on finite-dimensional spaces completely fail in the infinite-dimensional case: for instance, on any Banach space isomorphic to its Cartesian square and for any natural number n _ 2 there exists a C1-diffeomorphism of pure period n with no fixed point. This work opens the way to several interesting open questions on nonseparable Banach spaces: Does every Banach space with a C1 smooth norm admit a nonequivalent C1-smooth norm? In which Banach spaces is every compact subset the set where a certain C1 real-valued function vanishes?