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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces


Authors: Daniel Azagra, Robb Fry and Alejandro Montesinos
Journal: Proc. Amer. Math. Soc. 133 (2005), 727-734
MSC (2000): Primary 46B20
Published electronically: October 21, 2004
MathSciNet review: 2113921
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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 $C^p$-smooth Lipschitz bump functions, and $U$ is a bounded open subset of $X$, then, for every uniformly continuous function $f:Y\cap U\to\mathbb{R}$ and every $\varepsilon>0$, there exists a $C^p$-smooth Lipschitz function $F:X\to\mathbb{R}$ such that $\vert F(y)-f(y)\vert\leq\varepsilon$ for every $y\in Y\cap U$.


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Additional Information

Daniel Azagra
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Email: daniel_azagra@mat.ucm.es

Robb Fry
Affiliation: Department of Mathematics and Computer Science, St. Francis Xavier University, P.O. Box 5000, Antigonish, Nova Scotia, Canada B2G 2W5
Email: rfry@stfx.ca

Alejandro Montesinos
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Email: a_montesinos@mat.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07715-9
PII: S 0002-9939(04)07715-9
Received by editor(s): January 26, 2003
Published electronically: October 21, 2004
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.