Tensor Representation of Spaces of Holomorphic Functions and Applications

Abstract

In this paper we study the tensor representation of spaces of Frechet-valued holomorphic functions \([H(U, F),\tau ]\) in the form \( [(H(U), \tau )] \widehat{\otimes }_{\pi }F\) where U is an open subset of a Frechet space and \(\tau \in \{\tau _0, \tau _\omega , \tau _\delta \}.\) Using this result we consider the following problems: exponential laws for the topologies \(\tau _0, \tau _\omega \) on the space \(H(U \times V)\) where U and V are two open subsets of locally convex spaces E and F respectively; the coincidence of the topologies \(\tau _0, \tau _\omega , \tau _\delta \) on spaces of locally convex valued holomorphic functions (resp. germs) H(U, F) [resp. H(K, F)]; the inheritance of the properties (QNo), \((QNo)'\) via the spaces of holomorphic functions and of holomorphic germs.