Lectures on Vector Spaces of Analytic Functions

Abstract

These lectures deal with the field of interaction between the theory of analytic functions and functional analysis. In the first section, we shall study the classical theory of analytic functions of a single complex variable, using functional analysis as a tool to obtain some basic classical theorems like Cauchy’s integral theorem and the Mittag-Leffler theorem. In the second section, we dualize a gap theorem of C. Renyi for periodic entire functions to obtain some approximation theorems. One of these is a theorem about weighted polynomial approximation on the integers. In the third section we make a new application of perturbation of a basis in a Banach space to prove uniqueness theorems for analytic functions of one and of several complex variables. In the fourth section, we sketch a theory of duality of vector spaces of entire functions that leads to the solution of the problem of spectral synthesis for mean-periodic entire functions of one complex variable. The remainder of the lectures is devoted to a study of the space of bounded analytic functions in various weak topologies.