Spaces of Vector Valued Real Analytic Functions

Abstract

In this paper we study the spaces of weakly and strongly analytic functions with values in a locally convex topological vector space F and we look for conditions on F such that these two spaces (which are different in general) should coincide. In the case of vector valued C' functions and of vector valued holomorphic functions, Grothendieck proved (cf. [4; 7]) that it suffices to assume F complete (even less, quasi-complete, i.e., closed bounded sets are complete) to conclude that the two notions of weakly C0 (resp. weakly holomorphic) functions and strongly CO (resp. holomorphic) functions coincide. As we show with an example (cf. ?2) the sole condition of completeness of F does not imply strong analyticity from weak analyticity. On the other hand, it is known that if F is a Banach space then the two notions of real analyticity are the same [3]. For these reasons it is natural to raise the question of finding less restrictive conditions on F such that this occurs. The problem presents two aspects, one concerning the algebraic identification of the two spaces of analytic functions and the other the identification both in the algebraic and topological senses when these spaces are equipped with natural topologies. In order to deal with the algebraic case, we introduce the definition of quasi-(?9) spaces (cf. ?2, Definition 2) which generalizes the notion of (9.F) spaces introduced by Grothendieck in [5]. A quasi-(Q9) space still has one of the