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Twelve Landmarks of Twentieth-Century Analysis

Book Description

The striking theorems showcased in this book are among the most profound results of twentieth-century analysis. The authors' original approach combines rigorous mathematical proofs with commentary on the underlying ideas to provide a rich insight into these landmarks in mathematics. Results ranging from the proof of Littlewood's conjecture to the BanachÐTarski paradox have been selected for their mathematical beauty as well as educative value and historical role. Placing each theorem in historical perspective, the authors paint a coherent picture of modern analysis and its development, whilst maintaining mathematical rigour with the provision of complete proofs, alternative proofs, worked examples, and more than 150 exercises and solution hints. This edition extends the original French edition of 2009 with a new chapter on partitions, including the HardyÐRamanujan theorem, and a significant expansion of the existing chapter on the Corona problem.

Table of Contents

  1. Cover
  2. Half-title page
  3. Title page
  4. Copyright page
  5. Dedication
  6. Contents
  7. Foreword
  8. Preface
  9. 1. The Littlewood Tauberian theorem
    1. 1.1 Introduction
    2. 1.2 State of the art in 1911
    3. 1.3 Analysis of Littlewood’s 1911 article
    4. 1.4 Appendix: Power series
    5. Exercises
  10. 2. The Wiener Tauberian theorem
    1. 2.1 Introduction
    2. 2.2 A brief overview of Fourier transforms
    3. 2.3 Wiener’s original proof
    4. 2.4 Application to Littlewood’s theorem
    5. 2.5 Newman’s proof of the Wiener lemma
    6. 2.6 Proof of Wiener’s theorem using Gelfand theory
    7. Exercises
  11. 3. The Newman Tauberian theorem
    1. 3.1 Introduction
    2. 3.2 Newman’s lemma
    3. 3.3 The Newman Tauberian theorem
    4. 3.4 Applications
    5. 3.5 The theorems of Ikehara and Delange
    6. Exercises
  12. 4. Generic properties of derivative functions
    1. 4.1 Measure and category
    2. 4.2 Functions of Baire class one
    3. 4.3 The set of points of discontinuity of derivative functions
    4. 4.4 Differentiable functions that are nowhere monotonic
    5. Exercises
  13. 5. Probability theory and existence theorems
    1. 5.1 Introduction
    2. 5.2 Khintchine’s inequalities and applications
    3. 5.3 Hilbertian subspaces of L[sup(1)]([0,1])
    4. 5.4 Concentration of binomial distributions and applications
    5. Exercises
  14. 6. The Hausdorff–Banach–Tarski paradoxes
    1. 6.1 Introduction
    2. 6.2 Means
    3. 6.3 Paradoxes
    4. 6.4 Superamenability
    5. 6.5 Appendix: Topological vector spaces
    6. Exercises
  15. 7. Riemann’s “other” function
    1. 7.1 Introduction
    2. 7.2 Non-differentiability of the Riemann function at 0
    3. 7.3 Itatsu’s method
    4. 7.4 Non-differentiability at the irrational points
    5. Exercises
  16. 8. Partitio numerorum
    1. 8.1 Introduction
    2. 8.2 The generating function
    3. 8.3 The Dedekind η function
    4. 8.4 An equivalent of p(n)
    5. 8.5 The circle method
    6. 8.6 Asymptotic developments and numerical calculations
    7. 8.7 Appendix: Calculation of an integral
    8. Exercises
  17. 9. The approximate functional equation of the function θ[sub(0)]
    1. 9.1 The approximate functional equation
    2. 9.2 Other forms of the approximate functional equation and applications
    3. Exercises
  18. 10. The Littlewood conjecture
    1. 10.1 Introduction
    2. 10.2 Properties of the L[sup(1)]-norm and the Littlewood conjecture
    3. 10.3 Solution of the Littlewood conjecture
    4. 10.4 Extension to the case of real frequencies
    5. Exercises
  19. 11. Banach algebras
    1. 11.1 Spectrum of an element in a Banach algebra
    2. 11.2 Characters of a Banach algebra
    3. 11.3 Examples
    4. 11.4 C*-algebras
    5. Exercises
  20. 12. The Carleson corona theorem
    1. 12.1 Introduction
    2. 12.2 Prerequisites
    3. 12.3 Beurling’s theorem
    4. 12.4 The Lagrange–Carleson problem for an infinite sequence
    5. 12.5 Applications to functional analysis
    6. 12.6 Solution of the corona problem
    7. 12.7 Carleson’s initial proof and Carleson measures
    8. 12.8 Extensions of the corona theorem
    9. Exercises
  21. 13. The problem of complementation in Banach spaces
    1. 13.1 Introduction
    2. 13.2 The problem of complementation
    3. 13.3 Solution of problem (9)
    4. 13.4 The Kadeč–Snobar theorem
    5. 13.5 An example “à la Liouville”
    6. 13.6 An example “à la Hermite”
    7. 13.7 More recent developments
    8. Exercises
  22. 14. Hints for solutions
    1. Exercises for Chapter 1
    2. Exercises for Chapter 2
    3. Exercises for Chapter 3
    4. Exercises for Chapter 4
    5. Exercises for Chapter 5
    6. Exercises for Chapter 6
    7. Exercises for Chapter 7
    8. Exercises for Chapter 8
    9. Exercises for Chapter 9
    10. Exercises for Chapter 10
    11. Exercises for Chapter 11
    12. Exercises for Chapter 12
    13. Exercises for Chapter 13
  23. References
  24. Notations
  25. Index