 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 133, Number 3, Pages 727734
S 00029939(04)077159
Article electronically published on October 21, 2004
PERTURBED SMOOTH LIPSCHITZ EXTENSIONS
OF UNIFORMLY CONTINUOUS FUNCTIONS
ON BANACH SPACES
DANIEL AZAGRA, ROBB FRY, AND ALEJANDRO MONTESINOS
(Communicated by Jonathan M. Borwein)
Abstract. We show that if Y is a separable subspace of a Banach space X
such that both X and the quotient X/Y have Cpsmooth Lipschitz bump
functions, and U is a bounded open subset of X, then, for every uniformly
continuous function f : Y U R and every > 0, there exists a Cpsmooth
Lipschitz function F : X R such that F (y)  f(y) for every y Y U.
If we are given a separable subspace Y of a Banach space X and a continuous
(resp. Lipschitz) function f : Y R, under what conditions can we ensure the
existence of a Cp
smooth (Lipschitz) perturbed extension of f? That is, for a given
> 0, does there exist a Cp
