This textbook provides an introduction to modern analysis aimed at advanced undergraduate and graduate-level students of mathematics. Professional academics will also find this to be a useful reference work. It covers measure theory, basic functional analysis, single operator theory, spectral theory of bounded and unbounded operators, semigroups of operators, and Banach algebras. Further, this new edition of the textbook also delves deeper into C*-algebras and their standard constructions, von Neumann algebras, probability and mathematical statistics, and partial differential equations.
Most chapters contain relatively advanced topics alongside simpler ones, starting from the very basics of modern analysis and slowly advancing to more involved topics. The text is supplemented by many exercises, to allow readers to test their understanding and practical analysis skills.
Author(s): Shmuel Kantorovitz, Ami Viselter
Series: Oxford Graduate Texts in Mathematics
Edition: 2
Publisher: Oxford University Press
Year: 2022
Language: English
Pages: 592
Tags: Measure Theory, Linear Functionals, Bounded Operators, Banach Algebras, Hilbert Spaces, Unbounded Operators, C*-Algebras, Von Neumann Algebras, Distributions
Cover
Titlepage
Copyright
Dedication
Contents
Preface to the First Edition
Preface to the Second Edition
1 Measures
1.1 Measurable sets and functions
1.2 Positive measures
1.3 Integration of non-negative measurable functions
1.4 Integrable functions
1.5 Lp-spaces
1.6 Inner product
1.7 Hilbert space: a first look
1.8 The Lebesgue–Radon–Nikodym theorem
1.9 Complex measures
1.10 Convergence
1.11 Convergence on finite measure space
1.12 Distribution function
1.13 Truncation
Exercises
2 Construction of measures
2.1 Semi-algebras
2.2 Outer measures
2.3 Extension of measures on algebras
2.4 Structure of measurable sets
2.5 Construction of Lebesgue–Stieltjes measures
2.6 Riemann vs. Lebesgue
2.7 Product measure
Exercises
3 Measure and topology
3.1 Partition of unity
3.2 Positive linear functionals
3.3 The Riesz–Markov representation theorem
3.4 Lusin's theorem
3.5 The support of a measure
3.6 Measures on Rk; differentiability
Exercises
4 Continuous linear functionals
4.1 Linear maps
4.2 The conjugates of Lebesgue spaces
4.3 The conjugate of Cc(X)
4.4 The Riesz representation theorem
4.5 Haar measure
Exercises
5 Duality
5.1 The Hahn–Banach theorem
5.2 Reflexivity
5.3 Separation
5.4 Topological vector spaces
5.5 Weak topologies
5.6 Extremal points
5.7 The Stone–Weierstrass theorem
5.8 Operators between Lebesgue spaces: Marcinkiewicz's interpolation theorem
5.9 Fixed points
5.10 The bounded weak*-topology
Exercises
6 Bounded operators
6.1 Category
6.2 The uniform boundedness theorem
6.3 The open mapping theorem
6.4 Graphs
6.5 Quotient space
6.6 Operator topologies
Exercises
7 Banach algebras
7.1 Basics
7.2 Commutative Banach algebras
7.3 Involutions and C*-algebras
7.4 Normal elements
7.5 The Arens products
Exercises
8 Hilbert spaces
8.1 Orthonormal sets
8.2 Projections
8.3 Orthonormal bases
8.4 Hilbert dimension
8.5 Isomorphism of Hilbert spaces
8.6 Direct sums
8.7 Canonical model
8.8 Tensor products
8.8.1 An interlude: tensor products of vector spaces
8.8.2 Tensor products of Hilbert spaces
Exercises
9 Integral representation
9.1 Spectral measure on a Banach subspace
9.2 Integration
9.3 Case Z=X
9.4 The spectral theorem for normal operators
9.5 Parts of the spectrum
9.6 Spectral representation
9.7 Renorming method
9.8 Semi-simplicity space
9.9 Resolution of the identity on Z
9.10 Analytic operational calculus
9.11 Isolated points of the spectrum
9.12 Compact operators
Exercises
10 Unbounded operators
10.1 Basics
10.2 The Hilbert adjoint
10.3 The spectral theorem for unbounded selfadjoint operators
10.4 The operational calculus for unbounded selfadjoint operators
10.5 The semi-simplicity space for unbounded operators in Banach space
10.6 Symmetric operators in Hilbert space
10.7 Quadratic forms
Exercises
11 C*-algebras
11.1 Notation and examples
11.2 The continuous operational calculus continued
11.3 Positive elements
11.4 Approximate identities
11.5 Ideals
11.6 Positive linear functionals
11.7 Representations and the Gelfand–Naimark–Segal construction
11.7.1 Irreducible representations
11.8 Positive linear functionals and convexity
11.8.1 Pure states
11.8.2 Decompositions of functionals
Exercises
12 Von Neumann algebras
12.1 Preliminaries
12.2 Commutants
12.3 Density
12.4 The polar decomposition
12.5 W*-algebras
12.6 Hilbert–Schmidt and trace-class operators
12.7 Commutative von Neumann algebras
12.8 The enveloping von Neumann algebra of a C*-algebra
Exercises
13 Constructions of C*-algebras
13.1 Tensor products of C*-algebras
13.1.1 Tensor products of algebras
13.1.2 Tensor products of C*-algebras throughrepresentations
13.1.3 The maximal tensor product
13.1.4 Tensor products of bounded linear functionals
13.1.5 The minimal tensor product
13.1.6 Tensor products by commutative C*-algebras
13.2 Group C*-algebras
13.2.1 Unitary representations
13.2.2 The definition and representations of the group C*-algebra
13.2.3 Properties of the group C*-algebra
Exercises
Application I Probability
I.1 Heuristics
I.2 Probability space
I.2.1 L2-random variables
I.3 Probability distributions
I.4 Characteristic functions
I.5 Vector-valued random variables
I.6 Estimation and decision
I.6.1 Confidence intervals
I.6.2 Testing of hypothesis and decision
I.6.3 Tests based on a statistic
I.7 Conditional probability
I.7.1 Heuristics
I.7.2 Conditioning by an r.v.
I.8 Series of L2 random variables
I.9 Infinite divisibility
I.10 More on sequences of random variables
Application II Distributions
II.1 Preliminaries
II.2 Distributions
II.3 Temperate distributions
II.3.1 The spaces Wp,k
II.4 Fundamental solutions
II.5 Solution in E
II.6 Regularity of solutions
II.7 Variable coefficients
II.8 Convolution operators
II.9 Some holomorphic semigroups
Bibliography
Index
