© 2015

A Primer on Hilbert Space Theory

Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups


  • Provides a reader-friendly introduction to Hilbert space that will equip the reader for more demanding further study

  • Presents many clarifying examples

  • Builds a pedagogic bridge between the rigorous mathematical style and the more practical perspective of physicists


Part of the UNITEXT for Physics book series (UNITEXTPH)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Carlo Alabiso, Ittay Weiss
    Pages 1-21
  3. Carlo Alabiso, Ittay Weiss
    Pages 23-74
  4. Carlo Alabiso, Ittay Weiss
    Pages 75-111
  5. Carlo Alabiso, Ittay Weiss
    Pages 113-147
  6. Carlo Alabiso, Ittay Weiss
    Pages 149-198
  7. Carlo Alabiso, Ittay Weiss
    Pages 199-213
  8. Carlo Alabiso, Ittay Weiss
    Pages 215-249
  9. Back Matter
    Pages 251-255

About this book


This book is an introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, resides in the very high mathematical difficulty of even the simplest physical case. Within an ordinary graduate course in physics there is insufficient time to cover the theory of Hilbert spaces and operators, as well as distribution theory, with sufficient mathematical rigor. Compromises must be found between full rigor and practical use of the instruments. The book is based on the author's lessons on functional analysis for graduate students in physics. It will equip the reader to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude.

With respect to the original lectures, the mathematical flavor in all subjects has been enriched. Moreover, a brief introduction to topological groups has been added in addition to exercises and solved problems throughout the text. With these improvements, the book can be used in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.


Baire's theorem Banach fixed-point theorem Banach spaces Cauchy-Schwarz inequality Closed graph theorem Fixed-point techniques Fredholm equation Hahn-Banach theorem Hilbert spaces Linear spaces Metric spaces Normed spaces Open mapping theorem Semi-normed space Semimetric spaces Topological groups Topological spaces Uniform spaces Vector spaces Volterra equation

Authors and affiliations

  1. 1.Department of PhysicsUniversity of ParmaParmaItaly
  2. 2.South Pacific UniversitySuvaFiji Islands

About the authors

Prof. Carlo Alabiso obtained his Degree in Physics at Milan University and then taught at Parma University, Parma, Italy for more than 40 years (with a period spent as a research fellow at the Stanford Linear Accelerator Center and at Cern, Geneva). His teaching encompassed topics in quantum mechanics, special relativity, field theory, elementary particle physics, mathematical physics, and functional analysis. His research fields include mathematical physics (Pad\'{e} approximants), elementary particle physics (symmetries and quark models), and statistical physics (ergodic problems), and he has published articles in a wide range of national and international journals as well as the previous Springer book (with Alessandro Chiesa), Problemi di Meccanica Quantistica non Relativistica.

Dr. Ittay Weiss completed his BSc and MSc studies in Mathematics at the Hebrew University of Jerusalem and he obtained his PhD in mathematics from Universiteit Utrecht in the Netherlands. He spent an additional three years in Utrecht as an assistant professor of mathematics, teaching mathematics courses across the entire undergraduate spectrum both at Utrecht University and at the affiliated University College Utrecht. He is currently a mathematics lecturer at the University of the South Pacific. His research interests lie in the fields of algebraic topology and operad theory, as well as the mathematical foundations of analysis and generalizations of metric spaces. 

Bibliographic information

  • Book Title A Primer on Hilbert Space Theory
  • Book Subtitle Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups
  • Authors Carlo Alabiso
    Ittay Weiss
  • Series Title UNITEXT for Physics
  • Series Abbreviated Title UNITEXT for Physics
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Physics and Astronomy Physics and Astronomy (R0)
  • Hardcover ISBN 978-3-319-03712-7
  • Softcover ISBN 978-3-319-35351-7
  • eBook ISBN 978-3-319-03713-4
  • Series ISSN 2198-7882
  • Series E-ISSN 2198-7890
  • Edition Number 1
  • Number of Pages XVII, 255
  • Number of Illustrations 5 b/w illustrations, 0 illustrations in colour
  • Topics Mathematical Methods in Physics
    Functional Analysis
    Algebraic Topology
  • Buy this book on publisher's site
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“This book introduces essential ideas and results of Hilbert space and linear operator theory to beginners in quantum physics. … The textbook is suitable for advanced undergraduate or introductory graduate courses for both physics and mathematics students.” (Michael Frank, Mathematical Reviews, August, 2015)