C^∞ functions in infinite dimension and linear partial differential difference equations with constant coefficients

Abstract

This work is closed to [2] where a dense linear subspace $\mathbb{E}$(E) of the space ℰ(E) of the Silva C ^∞ functions on E is defined; the dual of $\mathbb{E}$(E) is described via the Fourier transform by a Paley-Wiener-Schwartz theorem which is formulated exactly in the same way as in the finite dimensional case. Here we prove existence and approximation result for solutions of linear partial differential difference equations in $\mathbb{E}$(E) with constant coefficients. We also obtain a Hahn-Banach type extension theorem for some C^∞ functions defined on a closed subspace of a DFN space, which is analogous to a Boland’s result in the holomorphic case [1].