Real analysis / Fon-Che Liu.

Format:

Book

Author/Creator:

Liu, Fon-Che, author.

Series:

Oxford graduate texts in mathematics ; 26.

Oxford graduate texts in mathematics ; 26

Language:

English

Subjects (All):

Mathematical analysis.

Physical Description:

viii, 310 pages ; 24 cm.

Edition:

First edition.

Place of Publication:

Oxford, United Kingdom : Oxford University Press, 2016.

Summary:

Real analysts is indispensable for in-depth understanding and effective application of methods of modern analysis. The core of the book describes the theory of functions of a real variable, framed in the setting of general measure and integration. Heavy emphasis is placed on measures and functions defined in Euclidean n-space; in particular, function spaces defined in terms of Lebesgue measure on Rn are treated in some detail including introduction of useful operations on these spaces, mindful that this area of real analysis plays a fundamental role in many mathematical fields and lends a helpful hand to analysis of various problems of mathematical physics and engineering. Numerous examples are included at each stage to illustrate the expressed ideas, with exercises scattered throughout the text. This book is intended primarily for graduate students, but can also be used in upper level undergraduate studies and for self-study. Book jacket.

Contents:

1 Introduction and Preliminaries 1

1.1 Summability of systems of real numbers 2

1.2 Double series 4

1.3 Coin tossing 6

1.4 Metric spaces and normed vector spaces 9

1.5 Semi-continuities 17

1.6 The space l^p(Z) 19

1.7 Compactness 24

1.8 Extension of continuous functions 35

1.9 Connectedness 36

1.10 Locally compact spaces 37

2 A Glimpse of Measure and Integration 40

2.1 Families of sets and set functions 40

2.2 Measurable spaces and measurable functions 43

2.3 Measure space and integration 47

2.4 Egoroff theorem and monotone convergence theorem 49

2.5 Concepts related to sets of measure zero 52

2.6 Fatou lemma and Lebesgue dominated convergence theorem 55

2.7 The space L^p(Δ, Σ μ 57

2.8 Miscellaneous remarks 61

3 Construction of Measures 65

3.1 Outer measures 65

3.2 Lebesgue outer measure on R 67

3.3 σ-algebra of measurable sets 70

3.4 Premeasures and outer measures 72

3.5 Carathéodory measures 80

3.6 Construction of Carathéodorymeasures 82

3.7 Lebesgue-Stieltjes measures 84

3.8 Borel regularity and Radon measures 88

3.9 Measure-theoretical approximation of sets in Rⁿ 89

3.10 Riesz measures 94

3.11 Existence of nonmeasurable sets 99

3.12 The axiom of choice and maximality principles 100

4 Functions of Real Variables 104

4.1 Lusin theorem 104

4.2 Riemann and Lebesgue integral 106

4.3 Push-forward of measures and distribution of functions 110

4.4 Functions of bounded variation 114

4.5 Riemann-Stieltjes integral 119

4.6 Covering theorems and differentiation 126

4.7 Differentiability of functions of a real variable and related functions 140

4.8 Product measures and Fubini theorem 150

4.9 Smoothing of functions 156

4.10 Change of variables for multiple integrals 160

4.11 Polar coordinates and potential integrals 168

5 Basic Principles of Linear Analysis 179

5.1 The Baire category theorem 179

5.2 The open mapping theorem 184

5.3 The closed graph theorem 185

5.4 Separation principles 187

5.5 Complex form of Hahn-Banach theorem 196

5.6 Hilbert space 198

5.7 Lebesgue-Nikodym theorem 204

5.8 Orthonormal families and separability 207

5.9 The space L²[-π, π] 211

5.10 Weak convergence 221

6 L^p Spaces 226

6.1 Some inequalities 226

6.2 Signed and complex measures 229

6.3 Linear functionals on L^p 242

6.4 Modular distribution function and Hardy-Littlewood maximal function 247

6.5 Convolution 252

6.6 The Sobolev space W^k,p(Δ) 258

7 Fourier Integral and Sobolev Space H^s265

7.1 Fourier integral for L¹ functions 265

7.2 Fourier integral on L² 274

7.3 The Sobolev space H^s 277

7.4 Weak solutions of the Poisson equation 280

7.5 Fourier integral of probability distributions 285.

Notes:

Includes bibliographical references (pages 301-302) and index.

ISBN:

0198790430

9780198790433

9780198790426

0198790422

OCLC:

958863120