A note on the range of the derivatives of analytic approximations of uniformly continuous functions on c0

[摘要] A real Banach space X satisfies property (K) (defined in. [M. Cepedello, P Hajek, Analytic approximations of uniformly continuous functions in real Banach spaces, J. Math. Anal. Appl. 256 (2001) 80-98]) if there exists a real-valued function on X which is uniformly (real) analytic and separating. We obtain that every uniformly continuous function f : U -> R. where U is an open subset of a separable Banach space X with property (K) and containing c(0) (thus X = c(0) circle plus Y for some Banach space Y) can be uniformly approximated by (real) analytic functions g : U -> R such that partial derivative g/partial derivative c(0) (U) subset of boolean AND(p>0) e(p) (where partial derivative f/partial derivative c(0) (U) is the set of partial derivatives {partial derivative f/partial derivative x (x. y): (x, y) is an element of U}). Similar statements are obtained for uniformly continuous functions f : U -> E with values in a (finite or infinite dimensional) Banach space E. Some consequences of these results are studied. (C) 2008 Elsevier Inc. All rights reserved.

[发布日期] 2008-12-15

[关键词] approximation by analytic functions;range of the derivatives