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

Abstract 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 ⊕ Y for some Banach space Y ) can be uniformly approximated by (real) analytic functions g : U → R such that ∂ g ∂ c 0 ( U ) ⊂ ⋂ p > 0 l p (where ∂ f ∂ c 0 ( U ) is the set of partial derivatives { ∂ f ∂ x ( x , y ) : ( x , y ) ∈ 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.