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Lectures on Functional Analysis and the Lebesgue Integral

  • Vilmos Komornik

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Functional Analysis

    1. Front Matter
      Pages 1-2
    2. Vilmos Komornik
      Pages 3-54
    3. Vilmos Komornik
      Pages 55-117
    4. Vilmos Komornik
      Pages 119-147
  3. The Lebesgue Integral

    1. Front Matter
      Pages 149-149
    2. Vilmos Komornik
      Pages 151-167
    3. Vilmos Komornik
      Pages 169-195
    4. Vilmos Komornik
      Pages 197-209
    5. Vilmos Komornik
      Pages 211-254
  4. Function Spaces

    1. Front Matter
      Pages 255-256
    2. Vilmos Komornik
      Pages 257-304
    3. Vilmos Komornik
      Pages 305-340
    4. Vilmos Komornik
      Pages 341-362
  5. Back Matter
    Pages 363-403

About this book

Introduction

This textbook, based on three series of lectures held by the author at the University of Strasbourg, presents functional analysis in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension. This approach leads naturally to the basic notions and theorems. Most results are illustrated by the small ℓp spaces. The Lebesgue integral, meanwhile, is treated via the direct approach of Frigyes Riesz, whose constructive definition of measurable functions leads to optimal, clear-cut versions of the classical theorems of Fubini-Tonelli and Radon-Nikodým.

Lectures on Functional Analysis and the Lebesgue Integral presents the most important topics for students, with short, elegant proofs. The exposition style follows the Hungarian mathematical tradition of Paul Erdős and others. The order of the first two parts, functional analysis and the Lebesgue integral, may be reversed. In the third and final part they are combined to study various spaces of continuous and integrable functions. Several beautiful, but almost forgotten, classical theorems are also included.

Both undergraduate and graduate students in pure and applied mathematics, physics and engineering will find this textbook useful. Only basic topological notions and results are used and various simple but pertinent examples and exercises illustrate the usefulness and optimality of most theorems. Many of these examples are new or difficult to localize in the literature, and the original sources of most notions and results are indicated to help the reader understand the genesis and development of the field.

Keywords

Functional analysis Hilbert space normed space Banach space dual space weak convergence reflexivity locally convex space Lebesgue integral spaces of continuous functions spaces of integrable functions

Authors and affiliations

  • Vilmos Komornik
    • 1
  1. 1.University of StrasbourgStrasbourgFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4471-6811-9
  • Copyright Information Springer-Verlag London 2016
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4471-6810-2
  • Online ISBN 978-1-4471-6811-9
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site