# Book:Peter D. Lax/Functional Analysis

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## Peter D. Lax:

## Peter D. Lax: *Functional Analysis*

Published $\text {2002}$, **Wiley**

- ISBN 978-0471556046.

### Subject Matter

### Contents

**Foreword**

**1. Linear Spaces**

- Axioms for linear spaces
- Infinite dimensional examples
- Subspace, linear span
- Quotient space
- Isomorphism
- Convex sets
- Extreme subsets

**2. Linear Maps**

- 2.1 Algebra of linear maps

- Axioms of linear maps
- Sums and composites
- Invertible linear maps
- Nullspace and range
- Invariant subspaces

- 2.2 Index of a linear map

- Degenrate maps
- Pseudoinverse
- Index
- Product formula for the index
- Stability of the index

**3. The Hahn-Banach Theorem**

- 3.1 The extension theorem

- Positive homogeneous, subadditive functionals
- Extension of linear functionals
- Gauge functions of convex sets

- 3.2 Geometric Hahn-Banach theorem

- The hyperplane separation theorem

- 3.3 Extensions of Hahn-Banach theorem

- The Agnew-Morse theorem
- The Bohnenblust-Sobczyk-Soukhomlinov theorem

**4. Applications of the Hahn-Banach Theorem**

- 4.1 Extension of positive linear functionals
- 4.2 Banach limits
- 4.3 Finitely additive invariant set functions

- Historical note

**5. Normed Linear Spaces**

- 5.1 Norms

- Norms for quotient spaces
- Complete normed linear spaces
- The spaces $C$, $B$
- $L^p$ spaces and Hölder's inequality
- Sobolev spaces, embedding theorems
- Separable spaces

- 5.2 Noncompactness of the unit ball

- Uniform convexity
- The Mazur-Ulam theorem on isometries

- 5.3 Isometries

**6. Hilbert Space**

- 6.1 Scalar product

- Schwarz inequality
- Parallelogram identity
- Completeness, closure
- $l^2,L^2$

- 6.2 Closest point in a closed convext subset

- Orthogonal complement of a subspace
- Orthogonal decomposition

- 6.3 Linear functionals

- The Riesz-Frechet representation theorem
- Lax-Milgram lemma

- 6.4 Linear span

- Orthogonal projection
- Orthonormal bases, Gram-Schmidt process
- Isometries of Hilbert space

**7. Applications of a Hilbert Space Results**

- 7.1 Radon-Nikodym theorem

- 7.2 Dirichlet's problem

- Use of the Riesz-Freceht theorem
- Use of the Lax-Milgram theorem
- Use of orthogonal decomposition

**8. Duals of Normed Linear Spaces**

- 8.1 Bounded linear functionals

- Dual space

- 8.2 Extension of bounded linear functionals

- Dual characterization of the norm
- Dual characterization of distance from a subspace
- Dual characterization of the closed linear span of a set

- 8.3 Reflexive spaces

- Reflexivity of $L^p$, $1 < p < \infty$
- Separable spaces
- Separability of the dual
- Dual of $\map C Q$, $Q$ compact
- Reflexivity of subspaces

- 8.4 Support function of a set

- Dual characterization of convex hull
- Dual characterization of distance from closed, convex set

**9. Applications of Duality**

- 9.1 Completeness of weighted powers

- 9.2 The Müntz approxiation theorem

- 9.3 Runge's theorem

- 9.4 Dual variational problems in function theory

- 9.5 Existence of Green's function

**10. Weak Convergence**

- 10.1 Uniform boundedness of weakly converging sequences

- Principle of uniform boundedness
- Weakly sequentially closed convex sets

- 10.2 Weak sequential compactness

- Compactness of unit ball in reflexive space

- 10.3 Weak* convergence

- Helly's theorem

**11. Applications of Weak Convergence**

- 11.1 Approximation of the $\delta$ function by continuous functions

- Toeplitz theorem on summability

- 11.2 Divergence of Fourier series

- 11.3 Approximate quadrature

- 11.4 Weak and strong analyticity of vector-valued functions

- 11.5 Existence of solutions of partial differential equations

- Galerkin's method

- 11.6 The representation of analytics functions with positive real part

- Herglotz-Riesz theorem

**12. The Weak and Weak* Topologies**

- Comparison with sequential topology
- Closed convex sets in the weak topology
- Weak compactness
- Alaoglu's theorem

**13. Locally Convex Topologies and the Krein-Milman Theorem**

- 13.1 Separation of points by linear functionals

- 13.2 The Krein-Milman theorem

- 13.3 The Stone-Weierstrass theorem

- 13.4 Choquet's theorem

**14. Examples of Convex Sets and Their Extreme Points**

- 14.1 Positive functionals

- 14.2 Convex functions

- 14.3 Completely monotone functions

- 14.4 Theorems of Caratheodory and Bochner

- 14.5 A theorem of Krein

- 14.6 Positive Harmonic functions

- 14.7 The Hamburger moment problem

- 14.8 G. Birkhoff's conjecture

- 14.9 De Finetti's theorem

- 14.10 Measure-preserving mappings

- Historical note

**15. Bounded Linear Maps**

- 15.1 Boundedness and continuity

- Norm of a bounded linear map
- Transpose

- 15.2 Strong and weak topologies

- Strong and weak sequential convergence

- 15.3 Principle of uniform boundedness

- 15.4 Composition of bounded maps

- 15.5 The open mapping principle

- Closed graph theorem

**16. Examples of Bounded Linear Maps**

- 16.1 Boundedness of integral operators

- Integral operators of Hilbert-Schmidt type
- Integral operators of Holmgren type

- 16.2 The convexity theorem of Marcel Riesz

- 16.3 Examples of bounded integral operators

- The Fourier transform, Parseval's theorem and Hausdorff-Young inequality
- The Hilbert transform
- The Laplace transform
- The Hilbert-Hankel transform

- 16.4 Solution operators for hyperbolic equations

- 16.5 Solution operator for the heat equation

- 16.6 Singular integral operators, pseudodifferential operators and Fourier integral operators

- Fourier integral operators

**17. Banach Algebras and their Elementary Spectral Theory**

- 17.1 Normed algebras

- Invertible elements
- Resolvent set and spectrum
- Resolvent
- Spectral radius

- 17.2 Functional calculus

- Spectral mapping theorem
- Projections

**18. Gelfand's Theory of Commutative Banach Algebras**

- Homomorphisms into $\C$
- Maximal ideals
- Mazur's lemma
- The spectrum as the range of homomorphisms
- The spectral mapping theorem revisited
- The Gelfand representation
- Gelfand topology

**19. Applications of Gelfand's Theory of Commutative Banach Algebras**

- 19.1 The algebra $\map C S$

- 19.2 Gelfand compactification

- 19.3 Absolutely convergent Fourier series

- 19.4 Analytic functions in the closed unit disk

- Analytic functions in the polydisk

- 19.5 Analytic functions in the open unit disk

- 19.6 Wiener's Tauberian theorem

- 19.7 Commutative $\mathcal B^*$ algebras

- Historical note

**20. Examples of Operators and Their Spectra**

- 20.1 Invertible maps

- Boundary points of the spectrum

- 20.2 Shifts

- 20.3 Volterra integral operators

- 20.4 The Fourier transform

**21. Compact Maps**

- 21.1 Basic properties of compact maps

- Compact maps form a two-sided ideal
- Identity plus compact map has index zero

- 21.2 The spectral theory of compact maps

- The transpose of a compact operator is compact
- The Fredholm alternative

- Historical note

**22. Examples of Compact Operators**

- 22.1 Compactness criteria

- Arela-Ascoli compactness criterium
- Rellich compactness criterium

- 22.2 Integral operators

- Hilbert-Schmidt operators

- 22.3 The inverse of elliptic partial differential operators

- 22.4 Operators defined by parabolic equations

- 22.5 Almost orthogonal bases

**23. Positive Compact Operators**

- 23.1 The spectrum of compact positive operators

- 23.2 Stochastic integral operators

- Invariant probability density

- 23.3 Inverse of a second order elliptic operator

**24. Fredholm's Theory of Integral Equations**

- 24.1 The Fredholm determinant and the Fredholm resolvent

- The spectrum of Fredholm operators
- A trace formula for Fredholm operators

- 24.2 Multiplicative property of the Fredholm determinant

- 24.3 The Gelfand-Levitan-Marchenko equation and Dyson's formula

**25. Invariant Subspaces**

- 25.1 Invariant subspaces of compact maps

- The von Neumann-Aronszajn-Smith theorem

- 25.2 Nested invariant subspaces

- Ringrose's theorem
- Unicelullar operators: the Brodsky-Donoghue theorem
- The Robinson-Bernstein and Lomonosov theorems
- Enflo's example

**26. Harmonic Analysis on a Halfline**

- 26.1 The Phragmén-Lindelöf principle for harmonic functions

- 26.2 An abstract Phragmén-Lindelöf principle

- Interior compactness

- 26.3 Asymptotic expansion

- Solutions of elliptic differential equation in a half-cylinder

**27. Index Theory**

- 27.1 The Noether index

- Pseudoinverse
- Stability of index
- Product formula
- Hörmander's stability theorem

- Historical note

- 27.2 Toeplitz operators

- Index-winding number
- The inversion of Toeplitz operators
- Discontinuous symbols
- Matrix Toeplitz operators

- 27.3 Hankel operators

**28. Compact Symmetric Operators in Hilbert Space**

- Variational principle for eigenvalues
- Completness for eigenfunctions
- The variational principles of Fisher and Courant
- Functional calculus
- Spectral theory of compact normal operators
- Unitary operators

**29. Examples of Compact Symmetric Operators**

- 29.1 Convolution

- 29.2 The inverse of a differential operator

- 29.3 The inverse of partial differential operators

**30. Trace Class and Trace Formula**

- 30.1 Polar decomposition and singular values

- 30.2 Trace class, trace norm, trace

- Matrix trace

- 30.3 The trace formula

- Weyl's inequalities
- Lidskii's theorem

- 30.4 The determinant

- 30.5 Examples and counterexamples of trace class operators

- Mercer's theorem
- The trace of integral operators
- A Volterra integral operator
- The trace of the powers of an operator

- 30.6 The Poisson summation formula

- Convolution of $S^1$ and the convergence of Fourier series
- The Selberg trace formula

- 30.7 How to express the index of an operator as a difference of traces

- 30.8 The Hilbert-Schmidt class

- Relation of Hilbert-Schmidt class and trace class

- 30.9 Determinant and trace for operator in Banach spaces

**31. Spectral Theory of Symmetric, Normal, and Unitary Operators**

- 31.1 The spectrum of symmetric operators

- Reality of spectrum
- Upper and lower bounds for the spectrum
- Spectral radius

- 31.2 Functional calculus for symmetric operators

- The square root of a positive operator
- Polar decomposition of bounded operators

- 31.3 Spectral resolution of symmetric operators

- Projection-valued measures

- 31.4 Absolutely continuous, singular and point spectra

- 31.5 The spectral representation of symmetric operators

- Spectral multiplicity
- Unitary equivalence

- 31.6 Spectral resolution of normal operators

- Functional calculus
- Commutative $B^*$ algebras

- 31.7 Spectral resolution of unitary operators

- Historical note

**32. Spectral Theory of Self-Adjoint Operators**

- The Hellinger-Toeplitz theorem
- Definition of self-adjointness
- Domain

- 32.1 Spectral resolution

- Sharpening of Herglotz's theorem
- Cauchy transform of measures
- The spectrum of self-adjoint operator
- Representation of the resolvent as a Cauchy transform
- Projection-valued measures

- 32.2 Spectral resolution using Cayley transform

- 32.3 A functional calculus for self-adjoint operators

**33. Examples of Self-Adjoint Operators**

- 33.1 The extension of unbounded symmetric operators

- 33.2 Examples of extension of symmetric operators; deficiency indices

- The operator $i \paren{\dfrac d {d x} }$ on $\map {C_0^1} \R$, $\map {C_0^1} {\R_+}$ and $C_0^1 \paren {0, 1}$
- Deficiency indices and von Neumann's theorem
- Symmetric operators in a real Hilbert space

- 33.3 The Friedrichs extension

- Semibounded symmetric operators
- Symmetric ODE
- Symmetric elliptic PDE

- 33.4 The Rellich perturbation theorem

- Self-adjointness of Schrodinger operators with singular potentials

- 33.5 The moment problem

- The Hamburger and Stieltjes moment problems
- Uniqueness, or not, of the moment problem

- Historical note

**34. Semigroups of Operators**

- 34.1 Strongly continuous one-parameter semigroups

- Infinitesimal generator
- Resolvent
- Laplace transform

- 34.2 The generation of semigroups

- The Hille-Yosida theorem

- 34.3 The approximation of semigroups

- The Lax equivalence theorem
- Trotter's product formula
- Strang's product formula

- 34.4 Perturbation of semigroups

- Lumer-Phillip's theorem
- Trotter's perturbation theorem

- 34.5 The spectral theory if semigroups

- Phillip's spectral mapping theorem
- Adjoint semigroups
- Semigroups of eventually compact operators

**35. Groups of Unitary Operators**

- 35.1 Stone's theorem

- Generation of unitary groups
- Positive definiteness and Bochner's theorem

- 35.2 Ergodic theory

- von Neumann's mean ergodic theorem

- 35.3 The Koopman group

- Volume-preserving flows
- Metric transitivity
- Time average
- Space average

- 35.4 The wave equation

- In full space-time
- In the exterior of an obstacle

- 35.5 Translation representation

- Sinai's theorem
- Incoming subspaces
- Solution of wave equation in odd number of space dimensions
- Wave propagation outside an obstacle

- 35.6 The Heisenberg commutation relation

- The uncertainty principle
- Weyl's form of the commutation relation
- von Neumman's theorem on pairs of operators that satisfy the commutation relation

- Historical note

**36. Examples of Strongly Continuous Semigroups**

- 36.1 Semigroups defined by parabolic equations

- 36.2 Semigroups defined by elliptic equations

- 36.3 Exponential decay of semigroups

- 36.4 The Lax-Phillips semigroup

- 36.5 The wave equation in the exterior of an obstacle

**37. Scattering Theory**

- 37.1 Perturbation theory

- 37.2 The wave operators

- 37.3 Existence of the wave operators

- 37.4 The invariance of wave operators

- 37.5 Potential scattering

- 37.6 The scattering operator

- Historical note

- 37.7 The Lax-Phillips scattering theory

- 37.8 The zeros of the scattering matrix

- 37.9 The automorphic wave equation

- Faddeev and Pavlor's theory
- The Riemann hypothesis

**38. A Theorem of Beurling**

- 38.1 The Hardy space

- 38.2 Beurling's theorem

- Inner and outer factors
- Factorization in the algebra of bounded analytic functions

- 38.3 The Titchmarsh convolution theorem

- Historical note

**Texts**

**A. A Riesz-Kakutani representation theorem**

- A.1 Positive linear functionals
- A.2 Volume
- A.3 $L$ as a space of functions
- A.4 Measurable sets and measure
- A.5 The Lebesgue measure and integral

**B. Theory of distributions**

- B.1 Definitions and examples
- B.2 Operations on distributions
- B.3 Local properties of distributions
- B.4 Applications to partial differential equations
- B.5 The Fourier transform
- B.6 Applications of the Fourier transform
- B.7 Fourier series

**C. Zorn's Lemma**

**Author Index**

**Subject Index**