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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 133, Number 3, Pages 727­734
S 0002-9939(04)07715-9
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 Cp-smooth 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 Cp-smooth
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

  

Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid

 

Collections: Mathematics