Smooth functions on c0

The space c0 lies at the heart of many constructions of higher order smooth functions on Banach spaces. To name a few, recall Torunczyk’s proof of the existence of C-smooth partitions of unity on WCG spaces ([12]) or Haydon’s recent constructions of C-bump functions on certain C(K) spaces ([7]). The crucial property of c0 that allows for those constructions is a rich supply of C -smooth functions that depend locally on finitely many coordinates. The main result of the present note (Theorem 6) implies that every C-smooth function on c0 has a locally compact derivative. This, in turn, means that every C-smooth function on c0 “almost” depends locally on finitely many coordinates, and confirms our intuition of c0 as being a very “flat” space. Our work was originally motivated by the question of Jaramillo (that we answer in the negative-see also [4]) whether there exists a C-smooth function on c0(Γ) which attains its minimum at exactly one point. However, our Corollaries (8-11) generalize to the case of C-smooth functions on c0 some results that Pelczynski [11] and Aron [1] obtained for polynomials and analytic functions on C(K) spaces, and some work of the author [6] on convex functions on c0. In particular, we show that every C-smooth (nonlinear) operator from c0 into a superreflexive space is locally compact. This implies that there exists no C-smooth operator from c0 onto l2, answering a question of S. Bates. The main idea of our work is contained in Lemma 2. Roughly speaking, it claims that a symmetric function with uniformly continuous derivative defined on c0 , with zero derivative at the origin, is almost constant along the basic vectors if n is large enough. Repeated applications of this principle lead to the proof of Theorem 6.