A Hilbert Space Problem Book pp 17-22 | Cite as

# Analytic Functions

Chapter

## 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 A^{2}(*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 A^{2}. No matter what*D*is, the set A^{2}(*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$$(f,g) = \int_D {f(z0g(z)*d\mu (z).} $$

## Keywords

Hilbert Space Analytic Function Kernel Function Hardy Space Bergman Kernel
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1982