Real Analysis

This textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry.

by Miklós Laczkovich (Author), Vera T. Sós (Author)
Includes insightful historical remarks regarding real analysis
Presents core ideas of analysis “as a way of thinking” as opposed to “a body of facts”
Explains how and why ideas arise, then how they evolve to the mature notions of real analysis

This textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry.

Authors: Loeb, Peter A
Written by one of the leading scholars in the field Includes a novel presentation of differentiation and absolute continuity using a local maximum function, resulting in an exposition that is both simpler and more general than the traditional approach
Theorems are stated for Lebesgue and Borel measures, with a note indicating when the same proof works only for Lebesgue measures
Appendices cover additional material, including theorems for higher dimensions and a short introduction to nonstandard analysis