Complex Function Theory for Functions with Arguments and Values in Locally Convex Linear Topological Spaces

Abstract

In Chapter I a brief introduction to the basic notions of locally convex linear topological spaces is given. In Chapter II, a theory of analytic functions is developed for functions of a complex variable with values in a sequentially complete locally convex complex linear topological space (l.t.s). The theory is sketched of continuous linear functions on one sequentially complete locally convex complex l.t.s. to a second such space. In the same spirit some theorems relating to functions of several complex variables taking their values in a sequentially complete locally convex complex l.t.s. are developed. In Chapter III, functions on one sequentially complete locally convex complex l.t.s to a second such space are studied and in particular notions of differentiability and analyticity. An analogue of the Cauchy-Riemann theory of functions of a complex variable is discussed.