Real Analysis

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.

1 - Introduction and Preliminaries
2 - A Glimpse of Measure and Integration
3 - Construction of Measures
4 - Functions of Real Variables
5 - Basic Principles of Linear Analysis
6 - Lp Spaces
7 - Fourier Integral and Sobolev Space Hs
8 - Postscript

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.