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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Characterization of Schwartz spaces by their holomorphic duals

Authors: Sten Bjon and Mikael Lindström
Journal: Proc. Amer. Math. Soc. 102 (1988), 909-913
MSC: Primary 46G20; Secondary 46A12
MathSciNet review: 934866
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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.

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Keywords: Schwartz spaces, holomorphic function, convergence structure
Article copyright: © Copyright 1988 American Mathematical Society