Approximation by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces

TL;DR

This paper discusses the approximation of uniformly continuous functions by smooth functions on certain Banach spaces, but identifies a gap in the previously claimed proof of such approximation results.

Contribution

It clarifies the conditions under which approximation by smooth functions is possible on weakly compactly generated Banach spaces, highlighting issues in earlier proofs.

Findings

Approximation results hold under specific conditions.

Identification of a gap in previous proof.

Clarification of approximation capabilities on Banach spaces.

Abstract

The following result was announced in the earlier version(s) of this paper: On weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded, real-valued functions by Lipschitz, C^{p} smooth functions. Unfortunately, there is a gap in the proof which renders the proof in its present form incorrect