Approximation by Cp-smooth, Lipschitz functions on Banach spaces

Abstract

If X is a Banach space with an unconditional basis and admits a C p -smooth, Lipschitz bump function, and Y is a subset of X, then any uniformly continuous function on Y can be uniformly approximated by functions which are C p -smooth and Lipschitz on X.  2005 Elsevier Inc. All rights reserved.

Corrigendum to “Approximation by Cp-smooth, Lipschitz functions on Banach spaces” [J. Math. Anal. Appl. 315 (2006) 599–605]

“…Nevertheless, using a construction from [2], techniques from [1], and employing a proof similar to that used originally, we are able to recover the results under the additional assumption that the subset Y ⊂ X is convex (see Theorem 1 below). For full details of the proof, we refer the reader to http://www.tru.ca/advtech/faculty/Robb_Fry.html, or via e-mail request (rfry@tru.ca).…”