© 2009

Partial Inner Product Spaces

Theory and Applications


Part of the Lecture Notes in Mathematics book series (LNM, volume 1986)

Table of contents

  1. Front Matter
    Pages I-XXIX
  2. Jean-Pierre Antoine, Camillo Trapani
    Pages 11-34
  3. Jean-Pierre Antoine, Camillo Trapani
    Pages 35-56
  4. Jean-Pierre Antoine, Camillo Trapani
    Pages 57-101
  5. Jean-Pierre Antoine, Camillo Trapani
    Pages 103-156
  6. Jean-Pierre Antoine, Camillo Trapani
    Pages 157-219
  7. Jean-Pierre Antoine, Camillo Trapani
    Pages 221-255
  8. Jean-Pierre Antoine, Camillo Trapani
    Pages 257-292
  9. Jean-Pierre Antoine, Camillo Trapani
    Pages 293-324
  10. Back Matter
    Pages 325-358

About this book


Partial Inner Product (PIP) Spaces are ubiquitous, e.g. Rigged Hilbert spaces, chains of Hilbert or Banach spaces (such as the Lebesgue spaces Lp over the real line), etc. In fact, most functional spaces used in (quantum) physics and in signal processing are of this type. The book contains a systematic analysis of PIP spaces and operators defined on them. Numerous examples are described in detail and a large bibliography is provided. Finally, the last chapters cover the many applications of PIP spaces in physics and in signal/image processing, respectively.
As such, the book will be useful both for researchers in mathematics and practitioners of these disciplines.


Chains of Banach or Hilbert spaces Hilbert space Partial *-algebras of operators Partial inner product spaces linear optimization

Authors and affiliations

  1. 1.Unité de Physique Théorique et deUniversité Catholique de LouvainLeuvenBelgium
  2. 2.Dipto. Matematica ed ApplicazioniUniversità di PalermoPalermoItaly

Bibliographic information

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From the reviews:

“Partial Inner Product (PIP) spaces generalize and synthesize a lot of spaces appearing in functional analysis, such as rigged Hilbert spaces, scales of Hilbert or Banach spaces, etc. … the book will be of  interest for researchers interested in function spaces as well as for those  interested in applications in theoretical physics and signal processing.” (Stefan Cobzaş, Zentralblatt MATH, Vol. 1195, 2010)

“The topic of this book is so-called PIP (partial inner product) spaces, which are vector spaces with a symmetric relation on pairs of elements … . Overall the book provides a unique opportunity for researchers working in the field of analysis to take a new perspective on more or less well known families of function spaces or operators, and the common functional analytic features common to all these families, analyzed in a very systematic way. … many readers will enjoy studying the proposed concepts.” (Hans G. Feichtinger, Mathematical Reviews, Issue 2011 i)