 
Summary: DELETING DIFFEOMORPHISMS WITH PRESCRIBED
SUPPORTS IN BANACH SPACES
DANIEL AZAGRA AND ALEJANDRO MONTESINOS
Abstract. We show that, for every infinitedimensional Banach space X with
a Schauder basis, the following are equivalent: (1) X has a Cp
smooth bump
function; (2) for every compact subset K and every open subset U of X with
K U, there exists a Cp
diffeomorphism h : X X \ K such that h is the
identity on X \ U.
A subset K of X is said to be topologically negligible provided there exists a home
omorphism h : X X \ K. The homeomorphism h is usually required to be the
identity outside a given neighborhood U of K. Here X can be a Banach space, a
manifold, or just a topological space, but we will only consider the case when X is
an infinitedimensional Banach space and h is a diffeomorphism (recall that points
are not topologically negligible in finitedimensional spaces). Such h will be called
a deleting diffeomorphism, and we will say that h has its support on U.
Deleting diffeomorphisms are very powerful tools in infinitedimensional global
analysis and nonlinear analysis. We do not intend to make a history of the develop
ment of topological negligibility and its applications, and we refer the reader to the
