Graduate Studies in Mathematics, Volume XXX, Amer. Math. Soc., Providence, (to appear).

Topics in Linear and Nonlinear Functional Analysis

Gerald Teschl

This manuscript provides a brief introduction to linear and nonlinear Functional Analysis. There is also an accompanying text on Real Analysis.

MSC: 46-01, 46E30, 47H10, 47H11, 58Exx, 76D05
Keywords: Functional Analysis, Banach space, Hilbert space, Mapping degree, fixed-point theorems, differential equations, Navier-Stokes equation

The text is available as pdf (2.3M) version. Any comments and bug reports are welcome!
Table of contents
  1. 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. Compacteness
    6. Bounded operators
    7. Sums and quotients of Banach spaces
    8. Spaces of continuous and differentiable functions
  2. Hilbert spaces
    1. Hilbert spaces
    2. The projection theorem and the Riesz lemma
    3. Operators defined via forms
    4. Orthogonal sums and tensor products
    5. Applications to Fourier series
  3. Compact operators
    1. Compact operators
    2. The spectral theorem for compact symmetric operators
    3. Applications to Sturm-Liouville operators
    4. Estimating eigenvalues
    5. Singular value decomposition of compact operators
    6. Hilbert-Schmidt and trace class operators
  4. The main theorems about Banach spaces
    1. The Baire theorem and its consequences
    2. The Hahn-Banach theorem and its consequences
    3. Refexivity
    4. The adjoint operator
    5. Weak convergence
  5. Bounded linear operators
    1. Banach algebras
    2. The C* algebra of operators and the spectral theorem
    3. Spectral measures
  6. More on convexity
    1. The geometric Hahn-Banach theorem
    2. Convex sets and the Krein-Milman theorem
    3. Weak topologies
    4. Beyond Banach spaces: Locally convex spaces
    5. Uniformly convex spaces
  7. Advanced spectral theory
    1. The Gelfand representation theorem
    2. Spectral theory for compact operators
    3. Fredholm operators
  8. Unbounded linear operators
    1. Closed operators
    2. Spectral theory for unbounded operators
    3. Reducing subspaces and spectral projections
    4. Relatively bounded and relatively compact operators
    5. Unbounded Fredholm operators
  9. Analysis in Banach spaces
    1. Single variable calculus in Banach spaces
    2. Multivariable calculus in Banach spaces
    3. Minimizing nonlinear functionals via calculus
    4. Minimizing nonlinear functionals II via compactness
    5. Contraction principles
    6. Ordinary differential equations
  10. 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
  11. 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
  12. Monotone operators
    1. Monotone operators
    2. The nonlinear Lax-Milgram theorem
    3. The main theorem of monotone operators
  13. Appendix: Some set theory
  14. Appendix: Metric and topological spaces
    1. Basics
    2. Convergence and completeness
    3. Functions
    4. Product topologies
    5. Compactness
    6. Connectedness
    7. Constructing continuous functions
    8. Initial and final topologies
    9. Continuous functions on metric spaces
Glossary of notations