Characterization of Schwartz Spaces by their Holomorphic Duals

Abstract

Let U be an open subset of a locally convex space E, and let Hc (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 HC{U,F) are relatively compact. Further it is shown that JE is a Schwartz space iff the continuous convergence structure on the algebra Ft(U) of scalar-valued holomorphic functions on {/, coincides with local uniform convergence. Using this, an example of a nuclear Frechet 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 HC(E) differs in this respect from that of its subspace LCE of linear forms and that of its superspace CC(E) of continuous functions. Introduction. In (11) H. Jarchow has proved that a locally convex space (les) E is Schwartz if and only if continuous convergence and local uniform convergence coincide on the dual of E. It is natural to ask if there is a holomorphic analogue of this result: Is a space E Schwartz if and only if continuous convergence and local uniform convergence coincide on a space H(U) of holomorphic functions on some open subset U of El We give a positive answer to this question using a result, closely related to Jarchow's, for continuous m-homogeneous polynomials (5). Further we prove that the space H(U) has the Montel property (bounded sets are relatively compact) when endowed with continuous convergence. For spaces of linear forms (on locally convex spaces) as well as for spaces of con- tinuous functions (on e.g. completely regular spaces) it is known that the locally convex topology associated with continuous convergence is the compact-open topol- ogy (cf. (1 and 10)). Using the above holomorphic characterization of Schwartz spaces, we provide an example which shows that spaces of holomorphic functions behave quite differently: There exists a nuclear Frechet space E, such that the locally convex topology associated with continuous convergence on H(E) is strictly finer than the compact-open topology. We recall some notation and definitions. All vector spaces in this paper are complex. A function f:U—*F into a convergence vector space (cvs) F is Gâteaux- holomorphic if the function A t—> I o f(x + Xh) is holomorphic in a neighborhood of zero for each x G U, h G E, and l G LF. It is holomorphic if it is Gâteaux- holomorphic and continuous. Let H(U, F) be the vector space of holomorphic