Topics in Real and Functional Analysis

Gerald Teschl

Abstract
This manuscript provides a brief introduction to Real and Functional Analysis. It covers basic Hilbert and Banach space theory as well as basic measure theory including Lebesgue spaces and the Fourier transform.

MSC: 46-01, 28-01, 46E30
Keywords: Functional Analysis, Banach space, Hilbert space, Measure theory, Lebesgue spaces, Fourier transform

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Table of contents
    Preface
  1. Introduction
    1. Linear partial differential equations
  2. A first look at Banach and Hilbert spaces
    1. Warm up: Metric and topological spaces
    2. The Banach space of continuous functions
    3. The geometry of Hilbert spaces
    4. Completeness
    5. Bounded operators
    6. Sums and quotients of Banach spaces
  3. Hilbert spaces
    1. Hilbert spaces
    2. The projection theorem and the Riesz lemma
    3. Operators defined via forms
    4. Orthogonal sums and tensor products
  4. Compact operators
    1. Compact operators
    2. The spectral theorem for compact symmetric operators
    3. Applications to Sturm-Liouville operators
  5. The main theorems about Banach spaces
    1. The Baire theorem and its consequences
    2. The Hahn-Banach theorem and its consequences
    3. Weak convergence
  6. More on Compact operators
    1. Canonical form of compact operators
    2. Hilbert-Schmidt and trace class operators
    3. Fredholm theory for compact operators
  7. Bounded linear operators
    1. Banach algebras
    2. The C* algebra of operators and the spectral theorem
    3. Spectral measures
    4. The Stone-Weierstraß theorem
  8. Almost everything about Lebesgue integration
    1. Borel measures in a nut shell
    2. Extending a premeasure to a measure
    3. Measurable functions
    4. Integration - Sum me up, Henri
    5. Product measures
    6. Transformation of measures and integrals
    7. Appendix: Transformation of Lebesgue--Stieltjes integrals
    8. Appendix: The connection with the Riemann integral
  9. The Lebesgue spaces Lp
    1. Functions almost everywhere
    2. Jensen ≤ Hölder ≤ Minkowski
    3. Nothing missing in Lp
    4. Approximation by nicer functions
    5. Integral operators
  10. More measure theory
    1. Decomposition of measures
    2. Derivatives of measures
    3. Complex measures
    4. Appendix: Functions of bounded variation and absolutely continuous functions
  11. The dual of Lp
    1. The dual of Lp, p<∞
    2. The dual of L∞ and the Riesz representation theorem
  12. The Fourier transform
    1. The Fourier transform on L1 and L2
    2. Interpolation and the Fourier transform on Lp
    3. Applications to linear partial differential equations
    4. The Marcinkiewicz interpolation theorem
    5. Sobolev spaces
  13. Appendix: Zorn's lemma

Bibliography
Glossary of notations
Index