 
Summary: C1
FINE APPROXIMATION OF FUNCTIONS ON BANACH
SPACES WITH UNCONDITIONAL BASIS
by DANIEL AZAGRA, JAVIER G ´OMEZ GIL§, JES ´US A. JARAMILLO¶,
MAURICIO LOVO
(Departamento de An´alisis Matem´atico, Facultad de Ciencias Matem´aticas, Universidad
Complutense, 28040 Madrid, Spain)
and ROBB FRY
(Department of Mathematics and Computer Science, St Francis Xavier University, PO Box
5000, Antigonish, Nova Scotia B2G 2W5, Canada)
[Received 16 September 2003]
Abstract
We show that if X is a Banach space having an unconditional basis and a C psmooth Lipschitz
bump function, then for every C1smooth function f from X into a Banach space Y, and for
every continuous function : X (0, ), there exists a C psmooth function g : X Y such
that f  g and f  g .
1. Introduction
Given a Fr´echet smooth function f between Banach spaces, we consider in this note the problem
of uniformly approximating both f and its derivative by functions with a higher order of
differentiability. More generally, if f : X Y is a Cksmooth function between Banach spaces,
