Functional Analysis

ISBN-10: 0471556041
ISBN-13: 9780471556046
Edition: 2002
Authors: Peter D. Lax
List price: $131.00 Buy it from $62.40
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Description: This advanced text on functional analysis is aimed at graduate students of functional analysis and professional mathematicians working in the fields of functional analysis and applied mathematics.

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Book details

List price: $131.00
Copyright year: 2002
Publisher: John Wiley & Sons, Incorporated
Publication date: 4/4/2002
Binding: Hardcover
Pages: 608
Size: 6.50" wide x 9.50" long x 1.50" tall
Weight: 2.090
Language: English

This advanced text on functional analysis is aimed at graduate students of functional analysis and professional mathematicians working in the fields of functional analysis and applied mathematics.

Foreword
Linear Spaces
Linear Maps
Algebra of linear maps
Index of a linear map
The Hahn-Banach Theorem
The extension theorem
Geometric Hahn-Banach theorem
Extensions of the Hahn-Banach theorem
Applications of the Hahn-Banach theorem
Extension of positive linear functionals
Banach limits
Finitely additive invariant set functions
Historical note
Normed Linear Spaces
Norms
Noncompactness of the unit ball
Isometries
Hilbert Space
Scalar product
Closest point in a closed convex subset
Linear functionals
Linear span
Applications of Hilbert Space Results
Radon-Nikodym theorem
Dirichlet's problem
Duals of Normed Linear Spaces
Bounded linear functionals
Extension of bounded linear functionals
Reflexive spaces
Support function of a set
Applications of Duality
Completeness of weighted powers
The Muntz approximation theorem
Runge's theorem
Dual variational problems in function theory
Existence of Green's function
Weak* Convergence
Uniform boundedness of weakly convergent sequences
Weak sequential compactness
Weak convergence
Applications of Weak Convergence
Approximation of the [delta] function by continuous functions
Divergence of Fourier series
Approximate quadrature
Weak and strong analyticity of vector-valued functions
Existence of solutions of partial differential equations
The representation of analytic functions with positive real part
The Weak and Weak* Topologies
Locally Convex Topologies and the Krein-Milman Theorem
Separation of points by linear functionals
The Krein-Milman theorem
The Stone-Weierstrass theorem
Choquet's theorem
Examples of Convex Sets and Their Extreme Points
Positive functionals
Convex functions
Completely monotone functions
Theorems of Caratheodory and Bochner
A theorem of Krein
Positive harmonic functions
The Hamburger moment problem
G. Birkhoff's conjecture
De Finetti's theorem
Measure-preserving mappings
Historical note
Bounded Linear Maps
Boundedness and continuity
Strong and weak topologies
Principle of uniform boundedness
Composition of bounded maps
The open mapping principle
Historical note
Examples of Bounded Linear Maps
Boundedness of integral operators
The convexity theorem of Marcel Riesz
Examples of bounded integral operators
Solution operators for hyperbolic equations
Solution operator for the heat equation
Singular integral operators, pseudodifferential operators and Fourier integral operators
Banach Algebras and their Elementary Spectral Theory
Normed algebras
Functional calculus
Gelfand's Theory of Commutative Banach Algebras
Applications of Gelfand's Theory of Commutative Banach Algebras
The algebra C(S)
Gelfand compactification
Absolutely convergent Fourier series
Analytic functions in the closed unit disk
Analytic functions in the open unit disk
Wiener's Tauberian theorem
Commutative B-algebras
Historical note
Examples of Operators and Their Spectra
Invertible maps
Shifts
Volterra integral operators
The Fourier transform
Compact Maps
Basic properties of compact maps
The spectral theory of compact maps
Historical note
Examples of Compact Operators
Compactness criteria
Integral operators
The inverse of elliptic partial differential operators
Operators defined by parabolic equations
Almost orthogonal bases
Positive compact operators
The spectrum of compact positive operators
Stochastic integral operators
Inverse of a second order elliptic operator
Fredholm's Theory of Integral Equations
The Fredholm determinant and the Fredholm resolvent
The multiplicative property of the Fredholm determinant
The Gelfand-Levitan-Marchenko equation and Dyson's formula
Invariant Subspaces
Invariant subspaces of compact maps
Nested invariant subspaces
Harmonic Analysis on a Halfline
The Phragmen-Lindelof principle for harmonic functions
An abstract Pragmen-Lindelof principle
Asymptotic expansion
Index Theory
The Noether index
Historical note
Toeplitz operators
Hankel operators
Compact Symmetric Operators in Hilbert Space
Examples of Compact Symmetric Operators
Convolution
The inverse of a differential operator
The inverse of partial differential operators
Trace Class and Trace Formula
Polar decomposition and singular values
Trace class, trace norm, and trace
The trace formula
The determinant
Examples and counterexamples of trace class operators
The Poisson summation formula
How to express the index of an operator as a difference of traces
The Hilbert-Schmidt class
Determinant and trace for operator in Banach spaces
Spectral Theory of Symmetric, Normal, and Unitary Operators
The spectrum of symmetric operators
Functional calculus for symmetric operators
Spectral resolution of symmetric operators
Absolutely continuous, singular, and point spectra
The spectral representation of symmetric operators
Spectral resolution of normal operators
Spectral resolution of unitary operators
Historical note
Spectral Theory of Self-Adjoint Operators
Spectral resolution
Spectral resolution using the Cayley transform
A functional calculus for self-adjoint operators
Examples of Self-Adjoint Operators
The extension of unbounded symmetric operators
Examples of the extension of symmetric operators; deficiency indices
The Friedrichs extension
The Rellich perturbation theorem
The moment problem
Historical note
Semigroups of Operators
Strongly continuous one-parameter semigroups
The generation of semigroups
The approximation of semigroups
Perturbation of semigroups
The spectral theory of semigroups
Groups of Unitary Operators
Stone's theorem
Ergodic theory
The Koopman group
The wave equation
Translation representation
The Heisenberg commutation relation
Historical note
Examples of Strongly Continuous Semigroups
Semigroups defined by parabolic equations
Semigroups defined by elliptic equations
Exponential decay of semigroups
The Lax-Phillips semigroup
The wave equation in the exterior of an obstacle
Scattering Theory
Perturbation theory
The wave operators
Existence of the wave operators
The invariance of wave operators
Potential scattering
The scattering operator
The Lax-Phillips scattering theory
The zeros of the scattering matrix
The automorphic wave equation
A Theorem of Beurling
The Hardy space
Beurling's theorem
The Titchmarsh convolution theorem
Historical note
Texts
Riesz-Kakutani representation theorem
Positive linear functionals
Volume
L as a space of functions
Measurable sets and measure
The Lebesgue measure and integral
Theory of distributions
Definitions and examples
Operations on distributions
Local properties of distributions
Applications to partial differential equations
The Fourier transform
Applications of the Fourier transform
Fourier series
Zorn's Lemma
Author Index
Subject Index

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