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On a problem of topologies in infinite dimensional holomorphy

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Ansemil, José María M. and Taskinen, J. (1990) On a problem of topologies in infinite dimensional holomorphy. Archiv der Mathematik, 54 (1). pp. 61-64. ISSN 0003-889X

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Abstract

The authors solve an interesting open problem concerning the equivalence of the compact-open topology τ0 and the Nachbin ported topology τω on spaces of holomorphic functions. (See, for example, the book by S. Dineen [Complex analysis in locally convex spaces, North-Holland, Amsterdam, 1981; MR0640093 (84b:46050)] for background.) Let H(U) denote the space of complex-valued holomorphic functions on an open subset U of a complex Fréchet-Montel space F. Ansemil and S. Ponte [Arch. Math. (Basel) 51 (1988), no. 1, 65–70; MR0954070 (90a:46109)] showed that these two topologies agree on H(U) for balanced U if and only if, for every natural number n, P(nF) is a Montel space. Using this result, they showed that for balanced open subsets U of certain non-Schwartz, Fréchet-Montel spaces, τ0=τω. Earlier, J. Mujica [J. Funct. Anal. 57 (1984), no. 1, 31–48; MR0744918 (86c:46050)] had shown that τ0=τω for Fréchet-Schwartz spaces. It is not hard to see that the two topologies differ if F is not Montel.
The authors' counterexample is the Fréchet-Montel space F of Taskinen [Studia Math. 91 (1988), no. 1, 17–30; MR0957282 (89k:46087)]. The authors observe that the complete symmetric projective tensor product Fs⊗ˆπF contains an isomorphic copy of l1. Consequently, P(2F) cannot be Montel, and the result follows.


Item Type:Article
Uncontrolled Keywords:Fréchet-Montel space; compact open topology; Nachbin topology
Subjects:Sciences > Mathematics > Topology
ID Code:16819
Deposited On:23 Oct 2012 08:04
Last Modified:09 Aug 2018 10:10

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