Daniel Alpay, An advanced complex analysis problem book. Topological vector spaces, functional analysis, and Hilbert spaces of analytic functions, Birkhäuser/Springer, New York, NY 2015, ix+525 p., ISBN 978-3-319-16058-0/pbk.

Abstract

Usually in a first course on Complex Analysis analytic functions are considered as individuals, not as elements of some Hilbert, Banach or Frechet spaces. Also some topological notions are introduced intuitively, without any rigorous topological foundation. Here one can mention the definition of the Riemann sphere bC simply as bC = C [ f1g, the introduction of the uniform convergence on compact sets via Morera's theorem, proofs of the Riemann mapping theorem without appealing to compactness arguments.