Abstract
Publisher Summary This chapter provides a proof of the result that is a pseudoconvex, finitely Runge open set U in a locally convex Hausdorff space with the approximation property is polynomially convex. An approximation theorem is also proved of the Runge type for such domains. A strong form of the Oka-Weil approximation theorem, which is useful in the study of the Nachbin topology on space of holomorphic functions on U is discussed in the chapter. A pseudoconvex, finitely Runge open set U subset of E is holomorphically convex. But in certain infinite dimensional spaces, for example in Silva spaces, every holomorphically convex domain is a domain of holomorphy or even a domain of existence.
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