Strong Nuclearity in Spaces of Holomorphic Mappings

Abstract

Publisher Summary This chapter presents a study on strong nuclearity in spaces of holomorphic mappings. Strong nuclearity of spaces of holornorphic are characterized, respectively Silva holomorphic, mappings on open subsets in locally convex and convex bornological, spaces. The space of all holomorphic functions on an arbitrary open subset of a strong dual of a strongly nuclear (F)-space is a strongly nuclear (F)-space under the compact open topology. All the proofs rely on the fact that a linear mapping between two normed spaces is strongly nuclear if and only if for every natural number “n” it can be represented as a composition of n nuclear mappings. The first proof consisted in a reduction to Boland's nuclearity result. The proof, which is presented, is a nice application of the nuclear bornology of a Banach space. Throughout the chapter, all locally convex (l.c) spaces are assumed to be Hausdorff and to be complex vector spaces, and a bornological vector (b.v.) space always denotes a complex, convex, separated, and complete bornological vector space in the terminology of Hogbe–Nlend.