Spaces of differentiable functions with the approximation property. (Spanish: Espacios de funciones diferenciables con la propiedad de aproximación).

Abstract

The author, in a joint paper with J. L. González Llvona [same journal 70 (1976), no. 4, 727–741; proved that a real Banach space E satisfies the approximation property if and only if the space Cnc(E), of n times continuously differentiable real functions, in the sense of Hadamard, on E, endowed with the topology that has the sets T(K,r)={f∈Cnc(E):Dpf(K)(Kp)⊂[−r,r],0≤p≤n} (where K runs over the compact sets of E and r>0) as a base for the neighborhoods of 0, satisfies the approximation property for some (and hence for every) n≥1. The author now proves this result when, instead of Cnc(E), one considers the space of n times continuously differentiable functions with respect to any other notion of differentiation which satisfies reasonable conditions (satisfied in particular by the Fréchet differential).