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Perturbed Smooth Lipschitz Extensions of Uniformly Continuous Functions on Banach Spaces

Daniel Azagra, Robb Fry and Alejandro Montesinos
Proceedings of the American Mathematical Society
Vol. 133, No. 3 (Mar., 2005), pp. 727-734
Stable URL: http://www.jstor.org/stable/4097746
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Perturbed Smooth Lipschitz Extensions of Uniformly Continuous Functions on Banach Spaces
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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 $f : Y \cap U \rightarrow \mathbb{R}$ and every $\varepsilon > 0$, there exists a $C^{p}-smooth$ Lipschitz function $F : X \rightarrow \mathbb{R}$ such that $\left \vert F(y) - f(y)\right \vert \leq \varepsilon$ for every $y \in Y \cap U$.

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