Analytic functions

Abstract

Analytic functions enter Hilbert space theory in several ways; one of their roles is to provide illuminating examples. The typical way to construct these examples is to consider a region D (“region” means a non-empty open connected subset of the complex plane), let μ be planar Lebesgue measure in D, and let A²(D) be the set of all complex-valued functions that are analytic throughout D and square-integrable with respect to μ. The most important special case is the one in which D is the open unit disc, D = {z: |z| < 1}; the corresponding function space will be denoted simply by A². No matter what D is, the set A²(D) is a vector space with respect to pointwise addition and scalar multiplication. It is also an inner-product space with respect to the inner product defined by

$$\left( {f,g} \right) = \int\limits_D {f\left( z \right)} g\left( z \right)*d\mu \left( z \right) $$

.