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Holomorphic approximation in infinite-dimensional Riemann domains. (English) Zbl 0584.32035

The main result of this paper is an infinite dimensional version of the classical Oka-Weil theorem on holomorphic approximation. Here is treated the case of pseudoconvex Riemann domains over a Fréchet space with basis. As an application it is proved that on the space of holomorphic functions on \(\Omega\), the compact-ported topology introduced by Nachbin coincides with the compact open topology, whenever \(\Omega\) is a pseudoconvex Riemann domain over a Fréchet-Schwartz space with basis. By passing to the envelope of holomorphy, this result is extended to the case of arbitrary Riemann domains over the same space. Using a result of Pełczyński, the existence of a basis in the previous results is replaced by the hypothesis of the bounded approximation property. The author obtains as particular cases some earlier results of M. Schottenloher [Ann. Inst. Fourier 26(1976), No.4, 207-237 (1976; Zbl 0309.32013)], J. A. Barroso [An. Acad. Bras. Cienc. 43, 527-546 (1971; Zbl 0246.46012)] etc.
Reviewer: I.Şerb

MSC:

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32T99 Pseudoconvex domains
32D10 Envelopes of holomorphy
32D05 Domains of holomorphy
32E05 Holomorphically convex complex spaces, reduction theory
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