Introduction to modern analysis.

Format:

Book

Author/Creator:

Kantorovitz, Shmuel, 1935- author.

Viselter, Ami, author.

Series:

Oxford graduate texts in mathematics ; 29.

Oxford scholarship online.

Oxford graduate texts in mathematics ; 29

Oxford scholarship online

Language:

English

Subjects (All):

Mathematical analysis.

Probabilities.

Physical Description:

1 online resource (593 pages)

Edition:

Second edition.

Place of Publication:

Oxford : Oxford University Press, [2022]

Summary:

This text is based on lectures given by the author in measure theory, functional analysis, Banach algebras, spectral theory (of bounded and unbounded operators), semigroups of operators, probability and mathematical statistics, and partial differential equations.

Contents:

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

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

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

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

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

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

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

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

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

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

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

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

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.

Notes:

This edition also issued in print: 2022.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

0-19-194465-3

0-19-266619-3