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E-grāmata: Real Analysis

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(Professor of Mathematics, National Taiwan University)
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Real Analysis is indispensable for in-depth understanding and effective application of methods of modern analysis. This concise and friendly book is written for early graduate students of mathematics or of related disciplines hoping to learn the basics of Real Analysis with reasonable ease. The essential role of Real Analysis in the construction of basic function spaces necessary for the application of Functional Analysis in many fields of scientific disciplines is demonstrated with due explanations and illuminating examples.

After the introductory chapter, a compact but precise treatment of general measure and integration is taken up so that readers have an overall view of the simple structure of the general theory before delving into special measures. The universality of the method of outer measure in the construction of measures is emphasized because it provides a unified way of looking for useful regularity properties of measures. The chapter on functions of real variables sits at the core of the book; it treats in detail properties of functions that are not only basic for understanding the general feature of functions but also relevant for the study of those function spaces which are important when application of functional analytical methods is in question. This is then followed naturally by an introductory chapter on basic principles of Functional Analysis which reveals, together with the last two chapters on the space of p-integrable functions and Fourier integral, the intimate interplay between Functional Analysis and Real Analysis. Applications of many of the topics discussed are included to motivate the readers for further related studies; these contain explorations towards probability theory and partial differential equations.

Recenzijas

The book is largely self-contained, including not only the basic concepts of measure theory that underlie integration, but also the linear analysis that provides the multidimensional spaces context... Recommended. * CHOICE * A good pace, many (very many) exercises, and a very nice presentation. Kudos to Prof. Liu. * Michael Berg, MAA Reviews *

Preface vii
1 Introduction and Preliminaries
1(39)
1.1 Summability of systems of real numbers
2(2)
1.2 Double series
4(2)
1.3 Coin tossing
6(3)
1.4 Metric spaces and normed vector spaces
9(8)
1.5 Semi-continuities
17(2)
1.6 The space lp(Z)
19(5)
1.7 Compactness
24(11)
1.8 Extension of continuous functions
35(1)
1.9 Connectedness
36(1)
1.10 Locally compact spaces
37(3)
2 A Glimpse of Measure and Integration
40(25)
2.1 Families of sets and set functions
40(3)
2.2 Measurable spaces and measurable functions
43(4)
2.3 Measure space and integration
47(2)
2.4 Egoroff theorem and monotone convergence theorem
49(3)
2.5 Concepts related to sets of measure zero
52(3)
2.6 Fatou lemma and Lebesgue dominated convergence theorem
55(2)
2.7 The space Lp(Ω, Σ, μ,)
57(4)
2.8 Miscellaneous remarks
61(4)
3 Construction of Measures
65(39)
3.1 Outer measures
65(2)
3.2 Lebesgue outer measure on R
67(3)
3.3 Σ-algebra of measurable sets
70(2)
3.4 Premeasures and outer measures
72(8)
3.5 Caratheodory measures
80(2)
3.6 Construction of Caratheodory measures
82(2)
3.7 Lebesgue--Stieltjes measures
84(4)
3.8 Borel regularity and Radon measures
88(1)
3.9 Measure-theoretical approximation of sets in Rn
89(5)
3.10 Riesz measures
94(5)
3.11 Existence of nonmeasurable sets
99(1)
3.12 The axiom of choice and maximality principles
100(4)
4 Functions of Real Variables
104(75)
4.1 Lusin theorem
104(2)
4.2 Riemann and Lebesgue integral
106(4)
4.3 Push-forward of measures and distribution of functions
110(4)
4.4 Functions of bounded variation
114(5)
4.5 Riemann-Stieltjes integral
119(7)
4.6 Covering theorems and differentiation
126(14)
4.7 Differentiability of functions of a real variable and related functions
140(10)
4.8 Product measures and Fubini theorem
150(6)
4.9 Smoothing of functions
156(4)
4.10 Change of variables for multiple integrals
160(8)
4.11 Polar coordinates and potential integrals
168(11)
5 Basic Principles of Linear Analysis
179(47)
5.1 The Baire category theorem
179(5)
5.2 The open mapping theorem
184(1)
5.3 The closed graph theorem
185(2)
5.4 Separation principles
187(9)
5.5 Complex form of Hahn--Banach theorem
196(2)
5.6 Hilbert space
198(6)
5.7 Lebesgue--Nikodym theorem
204(3)
5.8 Orthonormal families and separability
207(4)
5.9 The spaceL2[ -π, π]
211(10)
5.10 Weak convergence
221(5)
6 LP Spaces
226(39)
6.1 Some inequalities
226(3)
6.2 Signed and complex measures
229(13)
6.3 Linear functionals on Lp
242(5)
6.4 Modular distribution function and Hardy--Littlewood maximal function
247(5)
6.5 Convolution
252(6)
6.6 The Sobolev space Wk,p(Ω)
258(7)
7 Fourier Integral and Sobolev Space Hs
265(34)
7.1 Fourier integral for L1 functions
265(9)
7.2 Fourier integral on L2
274(3)
7.3 The Sobolev space Hs
277(3)
7.4 Weak solutions of the Poisson equation
280(5)
7.5 Fourier integral of probability distributions
285(14)
Postscript 299(2)
Bibliography 301(2)
List of Symbols 303(4)
Index 307
Fon-Che Liu received a first degree in Mathematics from Taiwan University (1962) and Ph.D. degree from Purdue University (1968). He joined the Institute of Mathematics, Academia Sinica in 1971 as Associate Research Fellow and was promoted to Research Fellow in 1973. He was Professor of Mathematics at Taiwan University between 1974 and 2005, and has been Visiting Professor at Purdue University and Wayne State University. Until 2000 he held the role of Director, Institute of Mathematics, Academia Sineca-Taipei, and was President of the Chinese Mathematical Society Taipei in the early 1990s.
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