Corrigendum to "Approximation by C^{p}-smooth, Lipschitz functions on Banach spaces" [J. Math. Anal. Appl., 315 (2006), 599-605]

TL;DR

This paper corrects a previous result by proving that in Banach spaces with an unconditional basis and smooth Lipschitz bump functions, uniformly continuous functions can be approximated by smooth Lipschitz functions, with controlled Lipschitz constants.

Contribution

It establishes a corrected approximation result for uniformly continuous functions on convex subsets of Banach spaces with specific smoothness and basis properties.

Findings

Uniform approximation by C^{p}-smooth Lipschitz functions in Banach spaces with unconditional basis.

Approximate Lipschitz constants depend only on the space, not on the functions.

Extension of approximation results to Banach space-valued functions.

Abstract

In this erratum, we recover the results from an earlier paper of the author's which contained a gap. Specifically, we prove that if X is a Banach space with an unconditional basis and admits a C^{p}-smooth, Lipschitz bump function, and Y is a convex subset of X, then any uniformly continuous function f: Y->R can be uniformly approximated by Lipschitz, C^{p}-smooth functions K:X->R. Also, if Z is any Banach space and f:X->Z is L-Lipschitz, then the approximates K:X->Z can be chosen CL-Lipschitz and C^{p}-smooth, for some constant C depending only on X.