Publication: On a problem of topologies in infinite dimensional holomorphy
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Publication Date
1990
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Birkhäuser Verlag
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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.
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J. Ansemil and S. Ponte, The compact open and the Nachbin ported topologies on spaces of holomorphic functions. Arch. Math.51, 65-70 (1988).
S.Dineen, Complex analysis in locally convex spaces. Math. Studies57, Amsterdam 1981.
G.Köthe, Topological vector spaces I-II. Berlin-Heidelberg-New York 1983 and 1979.
J. Mujica, A Banach-Dieudonné theorem for germs of holomorphic functions. J.Funct. Anal.(1)57, 34-48 (1984).
R. Ryan, Applications of topological tensor products to infinite dimensional holomorphy. Thesis, Trinity College, Dublin 1980.
J. Taskinen, The projective tensor product of Fréchet-Montel spaces. Studia Math.91, 17-30 (1988).