Holomorphic Mappings between Locally Convex Spaces

Abstract

In Section 3.1 we introduce Gâteaux holomorphic mappings between infinite dimensional spaces using only the classical concept of (ℂ-valued holomorphic function of one complex variable. This very weak definition leads to monomial and Taylor series expansions and connections with derivatives. Holomorphic (or Fréchet holomorphic) mappings are defined as continuous Gâteaux holomorphic mappings. With these basic definitions in place we give various examples of holomorphic functions and, in doing so, encounter new concepts such as bounding sets and uniform holomorphy that play a role in later developments. In Section 3.2 we consider holomorphic analogues of the topologies defined on spaces of homogeneous polynomials in Chapter 1.