The approximation property in spaces of differentiable functions. (Spanish: La propiedad de aproximación en espacios de funciones diferenciables).

Abstract

Let E and X be real Banach and locally convex spaces, respectively. Let Ccn(E,X) denote the space of n times continuously Hadamard differentiable functions f:E→X, endowed with the locally convex topology generated by the seminorms of the form f∈Ccn(E,X)→sup{α[Dpf(x)(y)]:x,y∈K}, where p∈N, p≤n, K⊂E is compact, and α is a continuous seminorm on X. The authors show that Ccn(E,X) is complete if X is complete, and investigate the relationship between approximation in Ccn(E,X) and the approximation property of E. For example, if X is complete, then Ccn(E,X) is topologically isomorphic to the ε-product of Ccn(E,R) and X. Using this result, the authors show that the following properties are equivalent: (a) E has the approximation property, (b) Ccn(E,R) has the approximation property for all (equivalently, for some) n≥1, and (c) for all X, Ccn(E,R)⊗X is dense in Ccn(E,X) for all (equivalently, for some) n≥1. Similar questions have been considered for the space Cn(E,X) of n times continuously Fréchet differentiable functions, endowed with the same locally convex topology. For example, the equivalence of (a) and (b) was proved by the first author (""Differentiable functions with the approximation property'', to appear) and the reviewer [Infinite dimensional holomorphy and applications (Proc. Internat. Sympos., Campinas, 1975), pp. 1–17, North-Holland, Amsterdam, 1977; see also Séminaire Pierre Lelong (Analyse), Année 1974/75, pp. 213–222, Lecture Notes in Math., Vol. 524, Springer, Berlin, 1976. It is apparently unknown whether Cn(E,R) is complete with this topology. Results relating the approximation property of E to approximation of Fréchet differentiable functions defined on open subsets of E and to generalized differentiable versions of the Stone-Weierstrass theorem have been obtained by J. B. Prolla and C. S. Guerreiro [Ark. Mat. 14 (1976), no. 2, 251–258].