Infinite dimensional holomorphy via categorical differential calculus

Abstract

Abstract. We establish that the category of holological spaces is equipped for calculus with complex scalars. This provides a theory of infinite dimensional holomorphy which allows maps to have nonconvex domains with empty interior. Some relatively elementary functions, hitherto excluded by the restrictive definitions of other theories, emerge as holomorphic maps.

A free convenient vector space for holomorphic spaces

“…Then we apply the Riemann mapping theorem and use the universal covering in order to get the general result. For a similar result for open convex subsets of C n, via the reflexiveness of the Fr6chet-Montel space H(U, C), see [8]. We obtain the category Hol (holomorphic spaces, holomorphic maps).…”

Infinite dimensional calculus allowing nonconvex domains with empty interior

“…Statements (B) and (C) of the next theorem each characterizes 'c#l-map on a primary domain' in the sense of [7]. So it shows equivalence of the new c#l-maps with those of [7] in all situations where primary domains are tractable, which is typically the case when N = R. See [6] for the case ~ = C. …”

“…In the case of c~ d, see [8] for the verifications; this category offers an alternative approach to the calculus of convenient vector spaces. The verifications for ~ will be found in [6].…”

“…This change has no visible significance in the case of real scalars: all non-degenerate real intervals are fundamental domains anyway in all the leading examples. But with complex scalars it turns out that the open f2 works for ~h while the more general A fails (this is explained in [6]). …”