Remarks on Germs in Infinite Dimensionsa

Smooth, real analytic and holomorphic mappings deened on non-open subsets of innnite dimensional vector spaces are treated. 0. Introduction In this paper we will generalize the concept of diierentiable maps f : E X ! F deened on open subsets to such on more general subsets of innnite dimensional vector spaces. We will refer to the theories for open domains as they have been developed in K82], K83] and F-K] for smooth (i.e. C 1) maps, in K-N] for holomorphic maps and in K-M] for real analytic maps. But before we start the general discussion, let us recall the nite dimensional situation for smooth maps. Let rst E = F = R and X be a non-trivial closed interval. Then a map f : X ! R is usually called smooth, if it is innnite often diierentiable on the interior of X and the one-sided derivatives of all orders exist. The later condition is equivalent to the condition, that all derivatives extend continuously from the interior of X to X. Furthermore, by Whitney's extension theorem (see W34]) these maps can also be described as being the restrictions to X of smooth maps on (some open neighborhood of X in) R. In case where X R is more general, these conditions fall apart. Now what happens if one changes to X R n. For closed convex sets with non-empty interior the corresponding conditions to the one dimensional situation still agree. In case of holomorphic and real analytic maps the germ on such a subset is already deened by the values on the subset. Hence we are actually speaking about germs in this situation. In innnite dimensions we will consider maps on just those convex subsets. So we do not claim greatest achievable generality, but rather restrict to a situation which is quite manageable. We will show that even in innnite dimensions the conditions above often coincide, and that real analytic and holomorphic maps on such sets are often germs of that class. Furthermore we have exponential laws for all three