Skip to main content
  • Textbook
  • © 1975

Geometric Functional Analysis and its Applications

Part of the book series: Graduate Texts in Mathematics (GTM, volume 24)

Buying options

eBook USD 69.99
Price excludes VAT (USA)
  • ISBN: 978-1-4684-9369-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 89.99
Price excludes VAT (USA)

This is a preview of subscription content, access via your institution.

Table of contents (4 chapters)

  1. Front Matter

    Pages i-x
  2. Convexity in Linear Spaces

    • Richard B. Holmes
    Pages 1-45
  3. Convexity in Linear Topological Spaces

    • Richard B. Holmes
    Pages 46-118
  4. Principles of Banach Spaces

    • Richard B. Holmes
    Pages 119-201
  5. Conjugate Spaces and Universal Spaces

    • Richard B. Holmes
    Pages 202-234
  6. Back Matter

    Pages 235-246

About this book

This book has evolved from my experience over the past decade in teaching and doing research in functional analysis and certain of its appli­ cations. These applications are to optimization theory in general and to best approximation theory in particular. The geometric nature of the subjects has greatly influenced the approach to functional analysis presented herein, especially its basis on the unifying concept of convexity. Most of the major theorems either concern or depend on properties of convex sets; the others generally pertain to conjugate spaces or compactness properties, both of which topics are important for the proper setting and resolution of optimization problems. In consequence, and in contrast to most other treatments of functional analysis, there is no discussion of spectral theory, and only the most basic and general properties of linear operators are established. Some of the theoretical highlights of the book are the Banach space theorems associated with the names of Dixmier, Krein, James, Smulian, Bishop-Phelps, Brondsted-Rockafellar, and Bessaga-Pelczynski. Prior to these (and others) we establish to two most important principles of geometric functional analysis: the extended Krein-Milman theorem and the Hahn­ Banach principle, the latter appearing in ten different but equivalent formula­ tions (some of which are optimality criteria for convex programs). In addition, a good deal of attention is paid to properties and characterizations of conjugate spaces, especially reflexive spaces.

Keywords

  • Banach Space
  • Convexity
  • calculus
  • compactness
  • functional analysis

Authors and Affiliations

  • Division of Mathematical Sciences, Purdue University, West Lafayette, USA

    Richard B. Holmes

Bibliographic Information

  • Book Title: Geometric Functional Analysis and its Applications

  • Authors: Richard B. Holmes

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4684-9369-6

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1975

  • Softcover ISBN: 978-1-4684-9371-9

  • eBook ISBN: 978-1-4684-9369-6

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: X, 246

  • Topics: Analysis

Buying options

eBook USD 69.99
Price excludes VAT (USA)
  • ISBN: 978-1-4684-9369-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 89.99
Price excludes VAT (USA)