Abstract
Publisher Summary This chapter discusses the fact that among the subjects studied in infinite dimensional holomorphy, the one of analytic continuation is perhaps the most developed. The Levi problem is positively answered in the case of Riemann domains over Banach spaces with Banach approximation property (B.A.P.). This result follows from investigations about properties of permanence for domains of existence under elementary set operations. It is observed that the behavior of the space of polynomials is directly related to the answer of the Levi problem. Holomorphic functions; morphism between Riemann domains over the same basic space; the concept of A-domain of holomorphy; etc., are all defined in the obvious way. A Banach space E is said to have the B.A.P., if E is separable and there exists a sequence of operators of finite rank.
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