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Smooth extension of functions on a certain class of non-separable Banach spaces

Jiménez Sevilla, María del Mar and Sánchez González, Luis (2011) Smooth extension of functions on a certain class of non-separable Banach spaces. Journal of Mathematical Analysis and Applications, 378 (1). pp. 173-183. ISSN 0022-247X

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Abstract

Let us consider a Banach space X with the property that every real-valued Lipschitz function f can be uniformly approximated by a Lipschitz, C1-smooth function g with Lip(g)⩽CLip(f) (with C depending only on the space X). This is the case for a Banach space X bi-Lipschitz homeomorphic to a subset of c0(Γ), for some set Γ, such that the coordinate functions of the homeomorphism are C1-smooth (Hájek and Johanis, 2010 . Then, we prove that for every closed subspace Y⊂X and every C1-smooth (Lipschitz) function f:Y→R, there is a C1-smooth (Lipschitz, respectively) extension of f to X. We also study C1-smooth extensions of real-valued functions defined on closed subsets of X. These results extend those given in Azagra et al. (2010) [4] to the class of non-separable Banach spaces satisfying the above property.


Item Type:Article
Uncontrolled Keywords:Smooth extensions; Smooth approximations
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:13817
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Deposited On:10 Nov 2011 12:54
Last Modified:06 Feb 2014 09:54

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