PYRAMIDAL VECTORS AND SMOOTH FUNCTIONS ON BANACH SPACES

Abstract. We prove that if X, Y are Banach spaces such that Y has non-trivial cotype and X has trivial cotype, then smooth functions from X into Y have a kind of \harmonic " behaviour. More precisely, we show that if is a bounded open subset of X and f:! Y is C1- smooth with uniformly continuous Frechet derivative, then f(@) is dense in f(). We also give a short proof of a recent result of P. Hajek. This note is motivated by recent results of P. Hajek ([H1], [H2]) concerning (Frechet) smooth nonlinear operators on the space c0. In [H1], Hajek proved (among other things) that if f: c0! R is a C1- smooth map with uniformly continuous derivative on Bc0, then f 0(Bc0) is a relatively com-pact subset of l1. From this, he deduced that if Y is a Banach space with non trivial type and f: c0! Y is C1- smooth with locally uniformly continuous derivative, then f is locally compact, which means that each point x 2 c0 has a neighbourhood V such that f(V) is relatively compact in Y. In [H2], he also proved that the same is true if Y has an unconditional basis and does not contain c0. These strikin