Approximation by \(C^p\)-smooth, Lipschitz functions on Banach spaces
The problem of uniformly approximating real continuous functions on Banach spaces by smooth functions has a long history. If the Banach space has a \(C^p\)-smooth bump function, \(C^p\)-smooth approximating functions are sought; the answer is positive, for example, in the class of weakly compactly generated Banach spaces [see, for example, \textit{R.\,Deville, G.\,Godefroy} and \textit{V.\,Zizler}, ``Smoothness and renormings in Banach spaces'' (Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman, Harlow and John Wiley \& Sons, New York) (1993; Zbl 0782.46019)]. If the bump function is, moreover, Lipschitz and the Banach space is separable, bounded uniformly continuous functions can be uniformly approximated by Lipschitz \(C^p\)-smooth functions [\textit{D.\,Azagra, R.\,Fry} and \textit{A.\,Montesinos}, Proc.\ Am.\ Math.\ Soc.\ 133, No.\,3, 727--734 (2005; Zbl 1060.46060)]. In this interesting note, answering in the positive a question in the aforesaid paper, independently posed by J.\,A.\,Jaramillo, the boundedness assumption is dropped when \(X\) has an unconditional Schauder basis. Precisely, the main theorem reads: Let \(X\) be a Banach space with an unconditional basis \(\{e_j,e^*_j\}_{j=1}^{\infty}\) which admits a \(C^p\)-smooth Lipschitz bump function. Let \(Y\subset X\) be a subset and \(f:Y\rightarrow \mathbb R\) a uniformly continuous map. Then for each \(\varepsilon>0\) there exists a \(C^p\)-smooth Lipschitz function \(g:X\rightarrow \mathbb R\) such that for all \(y\in Y\), \(| f(y)-g(y)| <\varepsilon\). Moreover, if \(X\) admits a \(C^p\)-smooth norm with unconditional constant \(C\), and if \(f\) is Lipschitz with constant \(\eta\), then we can choose \(g\) to have Lipschitz constant \(12C(1+C)\eta\). As a consequence, in a Banach space \(X\) with unconditional Schauder basis, it is equivalent to have a \(C^p\)-smooth Lipschitz bump function and the fact that for every subset \(Y\subset X\), uniformly continuous map \(f:Y\rightarrow \mathbb R\) and \(\varepsilon >0\), there exists a \(C^p\)-smooth Lipschitz map \(g:X\rightarrow \mathbb R\) with \(| f-g| <\varepsilon\) on \(Y\). Two natural questions are asked: the first one, if the unconditionality condition on the basis can be removed (or just if separability is enough); the second (suggested by D.\,Azagra), the optimality of the Lipschitz constant for \(g\) in the last part of the statement of the main theorem.
scientific article; zbMATH DE number 5016151