Linear Functional Analysis

Contents

Preface

Chapter 1  Introduction
     1.1. Topological spaces
          1.1.1. Topologies
          1.1.2. Compact spaces.
          1.1.3. Partitions of unity.
     1.2. Measure and integration
          1.2.1. Borel measures on a locally compact space X and positive linear forms on Cc(X)
          1.2.2. Complex measures
     1.3. Exercises
     References for further reading

Chapter 2  Normed spaces and operators
     2.1. Banach spaces
          2.1.1. Topological vector spaces
          2.1.2. Normed and Banach spaces.
          2.1.3. The space C(K) and the Stone-Weierstrass theorem
     2.2. Linear operators
          2.2.1. Bounded linear operators
          2.2.2. The space of bounded linear operators.
          2.2.3. Neumann series.
     2.3. Hilbert spaces
          2.3.1. Scalar products
          2.3.2. Orthogonal projections
          2.3.3. Orthonormal bases.
     2.4. Convolutions and summability kernels
          2.4.1. Integral operators
          2.4.2. Summability kernels on R^n
          2.4.3. Periodic summability kernels.
     2.5. The Riesz-Thorin interpolation theorem
     2.6. Applications to linear differential equations
          2.6.1. An initial value problem
          2.6.2. A boundary value problem
     2.7. Exercises
     References for further reading

Chapter 3  Frechet spaces and Banach theorems
     3.1. Frechet spaces
          3.1.1. Locally convex spaces.
          3.1.2. Frechet spaces.
     3.2. Banach theorems
          3.2.1. An application to the convergence problem of Fourier series
     3.3. Exercises
     References for further reading

Chapter 4  Duality
     4.1. The dual of a Hilbert space
          4.1.1. Riesz representation and Lax-Milgram theorem.
          4.1.2. The adjoint.
     4.2. Applications of the Riesz representation theorem
          4.2.1. Radon-Nikodym theorem
          4.2.2. The dual of L
          4.2.3. The dual of C(K).
     4.3. The Hahn-Banach theorem
          4.3.1. Analytic form of Hahn-Banach theorem
          4.3.2. The geometric Hahn-Banach theorem.
          4.3.3. Extension properties
          4.3.4. Proofs by duality: annihilators, total sets, completion, and the transpose
     4.4. Spectral theory of compact operators
          4.4.1. Elementary properties
          4.4.2. The Riesz-Fredholm theory
     4.5. Exercises
     References for further reading

Chapter 5  Weak topologies
     5.1. Weak convergence
     5.2. Weak and weak* topologies
     5.3. An application to the Dirichlet problem in the disc
     5.4. Exercises
     References for further reading

Chapter 6  Distributions
     6.1. Test functions
     6.2. The distributions
     6.3. Differentiation of distributions
     6.4. Convolution of distributions
          6.4.1. Support of a distribution and distributions with compact support.
          6.4.2. Convolution of distributions with functions
          6.4.3. Convolution of distributions.
     6.5. Distributional differential equations
          6.5.1. Linear differential equations.
          6.5.2. Fundamental solutions
          6.5.3. Green's functions.
          6.5.4. Green's function of the Dirichlet problem in the ball
     6.6. Exercises
     References for further reading

Chapter 7  Fourier transform and Sobolev spaces
     7.1. The Fourier integral
     7.2. The Schwartz class S
     7.3. Tempered distributions
          7.3.1. Fourier transform of tempered distributions.
          7.3.2. Plancherel Theorem.
     7.4. Fourier transform and signal theory
     7.5. The Dirichlet problem in the half-space
          7.5.1. The Poisson integral in the half-space.
          7.5.2. The Hilbert transform
     7.6. Sobolev spaces
          7.6.1. The spaces W^m,p
          7.6.2. The spaces H^m (R^n).
          7.6.3. The spaces H^m (\Omega)
          7.6.4. The spaces H(\Omega).
     7.7. Applications
          7.7.1. The Sturm-Liouville problem.
          7.7.2. The Dirichlet problem.
          7.7.3. Eigenvalues and eigenfunctions of the Laplacian
     7.8. Exercises
     References for further reading

Chapter 8  Banach algebras
     8.1. Definition and examples
     8.2. Spectrum
     8.3. Commutative Banach algebras
          8.3.1. Maximal ideals, characters, and the Gelfand transform
          8.3.2. Algebras of bounded analytic functions
     8.4. C*-algebras
          8.4.1. Involutions
          8.4.2. The Gelfand-Naimark theorem and functional calculus
     8.5. Spectral theory of bounded normal operators
          8.5.1. Functional calculus of normal operators
          8.5.2. Spectral measures
          8.5.3. Applications.
     8.6. Exercises
     References for further reading

Chapter 9  Unbounded operators in a Hilbert space
     9.1. Definitions and basic properties
          9.1.1. The adjoint
     9.2. Unbounded self-adjoint operators
          9.2.1. Self-adjoint operators
          9.2.2. Essentially self-adjoint operators.
          9.2.3. The Friedrichs extensions.
     9.3. Spectral representation of unbounded self-adjoint operators
     9.4. Unbounded operators in quantum mechanics
          9.4.1. Position, momentum, and energy
          9.4.2. States, observables, and Hamiltonian of a quantic system
          9.4.3. The Heisenberg uncertainty principle and compatible observables
          9.4.4. The harmonic oscillator.
     9.5. Appendix: Proof of the spectral theorem
          9.5.1. Functional calculus of a spectral measure
          9.5.2. Unbounded functions of bounded normal operators
          9.5.3. The Cayley transform.
          9.5.4. Proof of Theorem 9.20:
     9.6. Exercises
     References for further reading

Hints to exercises
     Chapter 1
     Chapter 2
     Chapter 3
     Chapter 4
     Chapter 5
     Chapter 6
     Chapter 7
     Chapter 8
     Chapter 9

Bibliography

Index