Topics in Real and Functional Analysis

Gerald Teschl

This manuscript provides a brief introduction to Real and (linear and nonlinear) 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, Mapping degree, fixed-point theorems, differential equations, Navier--Stokes equation

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Table of contents
  1. Warm up: Metric and topological spaces
    1. Basics
    2. Convergence
    3. Functions
    4. Product topologies
    5. Compactness
    6. Separation
  2. A first look at Banach and Hilbert spaces
    1. Introduction: Linear partial differential equations
    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
    4. Estimating eigenvalues
  5. The main theorems about Banach spaces
    1. The Baire theorem and its consequences
    2. The Hahn-Banach theorem and its consequences
    3. The adjoint operator
    4. The geometric Hahn-Banach theorem
    5. Weak convergence
    6. Weak topologies
    7. Beyond Banach spaces: Locally convex spaces
  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. Appendix: The connection with the Jordan content
  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. Hausdorff measure
    5. Infinite product measures
    6. Weak and vague convergence of measures
    7. 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. Applications to linear partial differential equations
    3. Sobolev spaces
    4. Applications to evolution equations
    5. Tempered distributions
  13. Interpolation
    1. Interpolation and the Fourier transform on Lp
    2. The Marcinkiewicz interpolation theorem
  14. Analysis in Banach spaces
    1. Differentiation and integration in Banach spaces
    2. Contraction principles
    3. Ordinary differential equations
  15. The Brouwer mapping degree
    1. Introduction
    2. Definition of the mapping degree and the determinant formula
    3. Extension of the determinant formula
    4. The Brouwer fixed point theorem
    5. Kakutani's fixed point theorem and applications to game theory
    6. Further properties and extensions
    7. The Jordan curve theorem
  16. The Leray-Schauder mapping degree
    1. The mapping degree on finite dimensional Banach spaces
    2. Compact operators
    3. The Leray-Schauder mapping degree
    4. The Leray-Schauder principle and the Schauder fixed point theorem
    5. Applications to integral and differential equations
  17. The stationary Navier-Stokes equation
    1. Introduction and motivation
    2. An insert on Sobolev spaces
    3. Existence and uniqueness of solutions
  18. Monotone operators
    1. Monotone operators
    2. The nonlinear Lax--Milgram theorem
    3. The main theorem of monotone operators
  19. Appendix: Some set theory
Glossary of notations