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Summary: ON DIFFEOMORPHISMS DELETING WEAK COMPACTA IN
BANACH SPACES
DANIEL AZAGRA AND ALEJANDRO MONTESINOS
Abstract. We prove that if X is an infinite-dimensional 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 non-reflexive Banach space
or an infinite-dimensional 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 infinite-dimensional 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 infinite-dimensional 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 infinite-dimensional Banach space. This discovery allowed him
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