 
Summary: ON DIFFEOMORPHISMS DELETING WEAK COMPACTA IN
BANACH SPACES
DANIEL AZAGRA AND ALEJANDRO MONTESINOS
Abstract. We prove that if X is an infinitedimensional Banach space with Cp
smooth partitions of the unity then X and X \ K are Cp
diffeomorphic, for every
weakly compact set K X.
1. Introduction, main results and preliminaries
In 1953 Victor L. Klee [20] proved that, if X is a nonreflexive Banach space
or an infinitedimensional Lp space and K is a compact subset of X, there exists
a homeomorphism between X and X \ K which is the identity outside a given
neighborhood of K. Klee also proved that for those infinitedimensional Banach
spaces X the unit sphere and the unit ball are homeomorphic to any of the closed
hyperplanes in X, and gave a topological classification of convex bodies in Hilbert
spaces. In subsequent papers, Bessaga and Klee generalized those results to ev
ery infinitedimensional normed space [8, 9, 12]. Klee's original proofs were of a
strong geometrical flavor: very beautiful, but rather difficult to handle in an ana
lytical way. Nevertheless, C. Bessaga found elegant explicit formulas for deleting
homeomorphisms, based on the existence of continuous noncomplete (nonequiva
lent) norms in every infinitedimensional Banach space. This discovery allowed him
