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Linear Functional Analysis

  • Textbook
  • © 2000

Overview

Authors:
  1. Bryan Patrick Rynne

    1. Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, UK

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    You can also search for this author in PubMed   Google Scholar

  2. Martin Alexander Youngson

    1. Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, UK

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • A new chapter on the Hahn-Banach theorem and extended material of the uniform boundedness theorem complete the coverage and make the book even more suitable for an introductory course on functional analysis
  • Detailed explanations and proofs and plenty of exercises - with full solutions provided at the back of the book - make this book ideal for reading courses and for self-study

Part of the book series: Springer Undergraduate Mathematics Series (SUMS)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-x
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  2. Preliminaries

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 1-30

  3. Normed Spaces

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 31-50

  4. Inner Product Spaces, Hilbert Spaces

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 51-85

  5. Linear Operators

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 87-121

  6. Linear Operators on Hilbert Spaces

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 123-160

  7. Compact Operators

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 161-189

  8. Integral and Differential Equations

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 191-219

  9. Solutions to Exercises

    • Bryan Patrick Rynne, Martin Alexander Youngson
    Pages 221-263

  10. Back Matter

    Pages 265-273
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Keywords

About this book

This book provides an introduction to the ideas and methods of linear func­ tional analysis at a level appropriate to the final year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the the­ ory of metric spaces). Part of the development of functional analysis can be traced to attempts to find a suitable framework in which to discuss differential and integral equa­ tions. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions defined on some set. In general, such a vector space is infinite-dimensional. This leads to difficulties in that, although many of the elementary properties of finite-dimensional vector spaces hold in infinite­ dimensional vector spaces, many others do not. For example, in general infinite­ dimensional vector spaces there is no framework in which to make sense of an­ alytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now briefly outline the contents of the book.

Reviews

From the reviews of the second edition:

"The authors write with a strong narrative thrust and a sensitive appreciation of the needs of the average student so that, by the final chapter, there is a real feeling of having "gotten somewhere worth getting" by a sensibly paced, clearly signposted route." Mathematical Gazette, 2000

"It is a fine book, with material well-organized and well-presented. A particularly useful feature is the material on compact operators and applications to differential equations." CHOICE magazine

"The presentation is quite elementary, and there are sufficiently many illuminating examples and exercises… this nice textbook perfectly fits the readership, i.e., undergraduate students in mathematics and physics… It may be recommended to all students who want to get in touch with the basic ideas of functional analysis and operator theory for the first time." Zentralblatt MATH

"This is an undergraduate introduction to functional analysis, with minimal prerequisites, namely linear algebra and some real analysis. … It is extensively cross-referenced, has a good index, a separate index of symbols (Very Good Feature), and complete solutions to all the exercises. It has numerous examples, and is especially good in giving both examples of objects that have a given property and objects that do not have the property." (Allen Stenger, MathDL, April, 2008)

"This second revised edition of the book … covers the normed aspects in functional analysis and consists of the preface, eight chapters, solutions to exercises (at the end of the book), a bibliography containing 17 references, notation index and subject index. … The book is readable and conceptually useful for undergraduate students in mathematics and physics. The authors show well how essential concepts from finite-dimensional linear algebra can be extended to the infinite-dimensional case." (Mohammad Sal Moslehian,Zentralblatt MATH, Vol. 1144, 2008)

Authors and Affiliations

  • Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, UK

    Bryan Patrick Rynne, Martin Alexander Youngson

Bibliographic Information

  • Book Title: Linear Functional Analysis

  • Authors: Bryan Patrick Rynne, Martin Alexander Youngson

  • Series Title: Springer Undergraduate Mathematics Series

  • DOI: https://doi.org/10.1007/978-1-4471-3655-2

  • Publisher: Springer London

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag London 2000

  • eBook ISBN: 978-1-4471-3655-2Published: 14 March 2013

  • Series ISSN: 1615-2085

  • Series E-ISSN: 2197-4144

  • Edition Number: 1

  • Number of Pages: X, 273

  • Topics: Analysis

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