Characterization of Schwartz spaces by their holomorphic duals

Abstract:

Let $U$ be an open subset of a locally convex space $E$, and let ${H_c}(U,F)$ denote the vector space of holomorphic functions into a locally convex space $F$, endowed with continuous convergence. It is shown that if $F$ is a semi-Montel space, then the bounded subsets of ${H_c}(U,F)$ are relatively compact. Further it is shown that $E$ is a Schwartz space iff the continuous convergence structure on the algebra $H(U)$ of scalar-valued holomorphic functions on $U$, coincides with local uniform convergence. Using this, an example of a nuclear Fréchet space $E$ is given, such that the locally convex topology associated with continuous convergence on $H(E)$ is strictly finer than the compact open topology. Thus, the behavior of the space ${H_c}(E)$ differs in this respect from that of its subspace ${L_c}E$ of linear forms and that of its superspace ${C_c}(E)$ of continuous functions.