© 2015

An Advanced Complex Analysis Problem Book

Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions


Table of contents

  1. Front Matter
    Pages i-xviii
  2. Analytic Functions

    1. Front Matter
      Pages 9-9
    2. Daniel Alpay
      Pages 11-60
    3. Daniel Alpay
      Pages 61-118
  3. Topology and Functional Analysis

    1. Front Matter
      Pages 119-119
    2. Daniel Alpay
      Pages 121-198
    3. Daniel Alpay
      Pages 199-247
    4. Daniel Alpay
      Pages 249-283
    5. Daniel Alpay
      Pages 285-327
  4. Hilbert Spaces of Analytic Functions

    1. Front Matter
      Pages 329-329
    2. Daniel Alpay
      Pages 405-441
    3. Daniel Alpay
      Pages 443-459
    4. Daniel Alpay
      Pages 461-474
    5. Daniel Alpay
      Pages 475-486
  5. Back Matter
    Pages 487-521

About this book


This is an exercises book at the beginning graduate level, whose aim is to illustrate some of the connections between functional analysis and the theory of functions of one variable. A key role is played by the notions of positive definite kernel and of reproducing kernel Hilbert space. A number of facts from functional analysis and topological vector spaces are surveyed. Then, various Hilbert spaces of analytic functions are studied.


Fourier transform Hilbert spaces of analytic functions complex variable reproducing kernel space topological vector spaces

Authors and affiliations

  1. 1.Department of MathematicsBen-Gurion University of the Negev Dept of Math, Bldg 37Beer-ShevaIsrael

About the authors

Prof. Daniel Alpay is a faculty member of the department of mathematics at Ben Gurion University, Beer Sheva, Israel. He is the incumbent of the Earl Katz Family chair in algebraic system theory. He has a double formation of electrical engineer (Telecom Paris, graduated 1978) and mathematician (PhD, Weizmann Institute, 1986). His research includes operator theory, stochastic analysis, and the theory of linear systems. Daniel Alpay is one of the initiators and responsible of the dual track electrical-engineering mathematics at Ben-Gurion University.

He is the author of "A Complex Analysis Problem Book" (Birkhäuser, 2011). Together with co-authors, he has written four books and more than 220 research papers, and edited twelve books of research papers.

Bibliographic information

  • Book Title An Advanced Complex Analysis Problem Book
  • Book Subtitle Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions
  • Authors Daniel Alpay
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-16058-0
  • eBook ISBN 978-3-319-16059-7
  • Edition Number 1
  • Number of Pages IX, 521
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Functional Analysis
    Operator Theory
  • Buy this book on publisher's site
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“The aim of this book is to fill in this gap, i.e. to get students familiar with some notions of functional analysis in the context of spaces of analytic functions, based on the unifying idea of reproducing kernel Hilbert space. … the book is dedicated to beginning graduate students aiming a specialization in complex analysis. Teachers of complex analysis will find some supplementary material here and those of functional analysis a source of concrete examples.” (S. Cobzaş, Studia Universitatis Babes-Bolyai, Mathematica, Vol. 61 (1), 2016)