Real Analysis 1st edition by Fon Che Liu – Ebook PDF Instant Download/Delivery: 0198790422 , 978-0198790426
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ISBN 10: 0198790422
ISBN 13: 978-0198790426
Author: Fon Che Liu
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.
Real Analysis 1st Table of contents:
1 Introduction and Preliminaries
1.1 Summability of systems of real numbers
1.2 Double series
1.3 Coin tossing
1.4 Metric spaces and normed vector spaces
1.5 Semi-continuities
1.6 The space lp (Z)
1.7 Compactness
1.8 Extension of continuous functions
1.9 Connectedness
1.10 Locally compact spaces
2 A Glimpse of Measure and Integration
2.1 Families of sets and set functions
2.2 Measurable spaces and measurable functions
2.3 Measure space and integration
2.4 Egoroff theorem and monotone convergence theorem
2.5 Concepts related to sets of measure zero
2.6 Fatou lemma and Lebesgue dominated convergence theorem
2.7 The space Lp (Ω,Σ,μ)
2.8 Miscellaneous remarks
3 Construction of Measures
3.1 Outer measures
3.2 Lebesgue outer measure on R
3.3 Σ-algebra of measurable sets
3.4 Premeasures and outer measures
3.5 Carathéodory measures
3.6 Construction of Carathéodory measures
3.7 Lebesgue–Stieltjes measures
3.8 Borel regularity and Radon measures
3.9 Measure-theoretical approximation of sets in Rn
3.10 Riesz measures
3.11 Existence of nonmeasurable sets
3.12 The axiom of choice and maximality principles
4 Functions of Real Variables
4.1 Lusin theorem
4.2 Riemann and Lebesgue integral
4.3 Push-forward of measures and distribution of functions
4.4 Functions of bounded variation
4.5 Riemann–Stieltjes integral
4.6 Covering theorems and differentiation
4.7 Differentiability of functions of a real variable and related functions
4.8 Product measures and Fubini theorem
4.9 Smoothing of functions
4.10 Change of variables for multiple integrals
4.11 Polar coordinates and potential integrals
5 Basic Principles of Linear Analysis
5.1 The Baire category theorem
5.2 The open mapping theorem
5.3 The closed graph theorem
5.4 Separation principles
5.5 Complex form of Hahn–Banach theorem
5.6 Hilbert space
5.7 Lebesgue–Nikodym theorem
5.8 Orthonormal families and separability
5.9 The space L2[–π, π]
5.10 Weak convergence
6 Lp Spaces
6.1 Some inequalities
6.2 Signed and complex measures
6.3 Linear functionals on Lp
6.4 Modular distribution function and Hardy–Littlewood maximal function
6.5 Convolution
6.6 The Sobolev space Wk,p (Ω)
7 Fourier Integral and Sobolev Space Hs
7.1 Fourier integral for L1 functions
7.2 Fourier integral on L2
7.3 The Sobolev space Hs
7.4 Weak solutions of the Poisson equation
7.5 Fourier integral of probability distributions
Postscript
Bibliography
List of Symbols
Index
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Tags: Fon Che Liu, Real Analysis