Real and functional analysis

Preface to the Book Series Moscow Lectures
Preface
Contents
CHAPTER 1 Metric and Topological Spaces
	1.1. Elements of Set Theory
	1.2. Metric Spaces
	1.3. Continuous Mappings
	1.4. The Contracting Mapping Principle
	1.5. Baire’s Category Theorem
	1.6. Topological Spaces
	1.7. Compact Sets and Their Properties
	1.8. Compactness Criteria
	1.9. Complements and Exercises
		1.9(i). Nets in topological spaces
		1.9(ii). Tychonoff’s theorem
		1.9(iii). Countable and sequential compactness
		1.9(iv). Functional separation of sets
		1.9(v). The Stone–Weierstrass theorem
		1.9(vi). The Cantor set
		1.9(vii). Cardinal characteristics of metric spaces
	Exercises
CHAPTER 2 Fundamentals of Measure Theory
	2.1. Introductory Remarks
	2.2. Algebras and -Algebras
	2.3. Additivity and Countable Additivity
	2.4. The Outer Measure and the Lebesgue Extension of Measures
	2.5. Lebesgue Measure and Lebesgue–Stieltjes Measures
	2.6. Signed Measures
	2.7. Complements and Exercises
		2.7(i). The Caratheodory measurability and extensions of measures
	Exercises
CHAPTER 3 The Lebesgue Integral
	3.1. Measurable Functions
	3.2. Convergence in Measure and Almost Everywhere
	3.3. The Construction of the Lebesgue Integral
	3.4. Passage to the Limit under the Integral Sign
	3.6. Criteria of Integrability
	3.7. Connections with the Riemann Integral
	3.8. The Hölder, Jensen and Minkowski Inequalities
	3.9. The Radon–Nikodym Theorem
	3.10. Products of Measure Spaces
	3.11. Fubini’s Theorem
	3.12. Complements and Exercises
		3.12(i). The Riemann integrability criterion
		3.12(ii). The image of a measure under a mapping
		3.12(iii). Uniform integrability
		3.12(iv). Liftings
	Exercises
CHAPTER 4 Connections between the Integral and Derivative
	4.1. Differentiable Functions
	4.2. Functions of Bounded Variation
	4.3. Absolutely Continuous Functions
	4.4. The Newton–Leibniz Formula
	4.5. Complements and Exercises
		4.5(i). Integration by parts in the Stieltjes integral
	4.5(ii). Convergence of Fourier series
	Exercises
CHAPTER 5 Normed and Euclidean Spaces
	5.1. Normed Spaces
	5.2. Examples
	5.3. Balls in Normed Spaces
	5.4. Orthonormal Systems, Bases, and Projections
	5.5. Convex Sets and the Schauder Theorem
	5.6. Complements and Exercises
		5.6(i). Balls and ellipsoids
		5.6(ii). The Kadec and Milyutin theorems
		5.6(iii). Ordered vector spaces and vector lattices
	Exercises
CHAPTER 6 Linear Operators and Functionals
	6.1. The Operator Norm and Continuity
	6.2. The Closed Graph Theorem
	6.3. The Hahn–Banach Theorem
	6.4. Applications of the Hahn–Banach Theorem
	6.5. Duals to Concrete Spaces
	6.6. The Weak and Weak-*Topologies
	6.7. Compactness in the Weak-*Topology
	6.8. Adjoint and Selfadjoint Operators
	6.9. Compact Operators
	6.10. Complements and Exercises
		6.10(i). Operator ranges and factorization
		6.10(ii). Weak compactness in Banach spaces
		6.10(iii). The Banach–Saks property and uniform convexity
		6.10(iv). Bases, approximations and complements
		6.10(v). Operators on ordered vector spaces
		6.10(vi). Vector integration
		6.10(vii). The Daniell integral
		6.10(viii). Interpolation theorems
	Exercises
CHAPTER 7 Spectral Theory
	7.1. The Spectrum of an Operator
	7.2. The Quadratic Form and Spectrum of a Selfadjoint Operator
	7.3. The Spectrum of a Compact Operator
	7.4. The Fredholm Alternative
	7.5. The Hilbert–Schmidt Theorem
	7.6. Unitary Operators
	7.7. Continuous Functions of Selfadjoint Operators
	7.8. The Functional Model
	7.9. Projections and Projection-Valued Measures
	7.10. Complements and Exercises
		7.10(i). The structure of the spectrum
		7.10(ii). Commuting selfadjoint operators
		7.10(iii). Operator ranges in a Hilbert space
		7.10(iv). Hilbert–Schmidt operators and nuclear operators
		7.10(v). Integral operators and Mercer’s theorem
		7.10(vi). Tensor products
		7.10(vii). Fredholm operators
		7.10(viii). The vector form of the spectral theorem
		7.10(ix). Invariant subspaces
	Exercises
CHAPTER 8 Locally Convex Spaces and Distributions
	8.1. Locally Convex Spaces
	8.2. Linear Mappings
	8.3. Separation of Convex Sets
	8.4. Distributions
	8.5. Derivatives of Distributions
	8.6. Complements and Exercises
		8.6(i). Metrizability and normability
		8.6(ii). The Mackey topology
		8.6(iii). Inductive and projective limits
		8.6(iv). Barrelled and bornological spaces
		8.6(v). Banach spaces generated by Minkowski functionals
		8.6(vi). The Krein–Milman theorem
		8.6(vii). The measurable graph theorem
		8.6(viii). Fix point theorems in locally convex spaces
	Exercises
CHAPTER 9 The Fourier Transform and Sobolev Spaces
	9.1. The Fourier Transform in L1
	9.2. The Fourier Transform in L2
	9.3. The Fourier Transform in S′
	9.4. Convolution
	9.5. The Spectrum of the Fourier Transform and Convolution
	9.6. The Laplace Transform
	9.7. Applications to Differential Equations
	9.8. Sobolev Spaces Wp,k
	9.9. Characterization of W2,k by Means of the Fourier Transform
	9.10. Complements and Exercises
		9.10(i). Singular integrals
		9.10(ii). Embedding theorems
		9.10(iii). The Bochner and Paley–Wiener theorems
	Exercises
CHAPTER 10 Unbounded Operators and Operator Semigroups
	10.1. Graphs and Adjoints
	10.2. Symmetric and Selfadjoint Operators
	10.3. The Spectral Theorem
	10.4. Unitary Invariants of Selfadjoint Operators
	10.5. Operator Semigroups
	10.6. Generators of Semigroups
	10.7. Complements and Exercises
		10.7(i). Extensions of symmetric operators
		10.7(ii). Semibounded forms and operators
		10.7(iii). The Chernoff and Trotter theorems
		10.7(iv). The mathematical model of quantum mechanics
		10.7(v). Sturm–Liouville operators
	Exercises
CHAPTER 11 Banach Algebras
	11.1. Basic Definitions
	11.2. Ideals
	11.3. Spectra
	11.4. Functional Calculus
	11.5. Commutative Banach Algebras
	11.6. The Structure of C*-Algebras
	11.7. Complements and Exercises
		11.7(i). The algebras C(K) and L
		11.7(ii). Von Neumann algebras
		11.7(iii). Haar measures and representations of groups
	Exercises
CHAPTER 12 Infinite-Dimensional Analysis
	12.1. Differentiability and Derivatives
	12.2. Properties of Differentiable Mappings
	12.3. Inverse and Implicit Functions
	12.4. Higher Order Derivatives
	12.5. Complements and Exercises
		12.5(i). Newton’s method
		12.5(ii). Multilinear mappings
		12.5(iii). Subdifferentials and monotone mappings
		12.5(iv). Approximations in Banach spaces
		12.5(v). Covering mappings
	Exercises
Comments
References
Index