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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.
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.
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