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Topological duality on the function space H(C^N)

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1979-01-01
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Elsevier
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By a classical theorem, there is an isomorphism between the space of entire functions of exponential type on Cn,ExpCn, and the analytic functions on H(Cn),H′(Cn) [see, for example, F. Trèves, Topological vector spaces, distributions, and kernels, Academic Press, New York, 1967; MR0225131 (37 #726)]. In this note, the author extends this useful theorem to H(CN), the space of analytic functions on the countable product of complex lines. Specifically, he considers H(CN) endowed with the compact-open topology τ0 and the associated bornological topology τδ. For both τ=τ0 and τδ, the author characterizes the strong duals (H(CN),τ)′ as spaces of entire functions of exponential type on CN. {Reviewer's remark: In the meantime the author has shown (private communication) that these dual spaces are different.}
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Análisis funcional y teoría de operadores
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J. BARROSO, Topologias nos espac;os de aplicages holomorfas entre espaqos localmente convexos, An. Acad. Brasil. Ci. 43 (1971), 527-546. J. BARROSO, Introducción a la holomorfia entre espacios normados. Cursos y Congresos de la Universidad de Santiago de Compostela, No. 7 (1976). S. DINEEN, Holomorphic functions on locally convex topological vector spaces, I, Ann. Inst. Fourier (Grenoble) 23 (1973), No. 1, 19-54. J. HORVATH, Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, Reading, Mass., 1966. J. M. ISIDRO, Topological duality on the function space (Zb(LT; F), Q), Proc. Royal Irish Acad. Sect. A, to appear. G. KöETHE, Topological Vector Spaces I, Springer-Verlag, New York/Berlin, 1969. L. NACHBIN, Holomorphic Functions, Domains of Holomorphy and Local Properties, North-Holland, Amsterdam. 1970. L. NACHBIN, Sur les espaces vectoriels topologiques d’applications continues, C. R.,4cad. Sci. Paris Ser. A 271 (1970), 596.598. I,. NACHBIN, Curso de Holomorfia entre EspaÇos Localmente Convexos, Universidad Federal do Rio de Janeiro, 1973. F. TREVES, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.
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