DiBenedetto E. Real Analysis

2nd Edition. — Springer Science+Business Media, New York, 2016. — 621 p. — ISBN: 1493940031This book is a self-contained introduction to real analysis assuming only basic notions on limits of sequences in RN, manipulations of series, their convergence criteria, advanced differential calculus, and basic algebra of sets.
The passage from the setting in RN to abstract spaces and their topologies is gradual. Continuous reference is made to the RN setting where most of the basic concepts originated.
The first eight chapters contain material forming the backbone of a basic training in real analysis. The remaining three chapters are more topical, relating to maximal functions, functions of bounded mean oscillation, rearrangements, potential theory and the theory of Sobolev functions. Even though the layout of the book is theoretical, the entire book and the last chapters in particular have in mind applications of mathematical analysis to models of physical phenomena through partial differential equations.
The preliminaries contain a review of the notions of countable sets and related examples. We introduce some special sets, such as the Cantor set and its variants and examine their structure. These sets will be a reference point for a number of examples and counterexamples in measure theory (Chapter 3) and in the Lebesgue differentiability theory of absolute continuous functions (Chapter 5). This initial Chapter contains a brief collection of the various notions of ordering, the Hausdorff maximal principle, Zorn’s Lemma, the well-ordering principle, and their fundamental connections.Preliminaries
Topologies and Metric Spaces
Measuring Sets
The Lebesgue Integral
Topics on Measurable Functions of Real Variables
The Lp Spaces
Banach Spaces
Spaces of Continuous Functions, Distributions, and Weak Derivatives
Topics on Integrable Functions of Real Variables
Embeddings of W1,p(E)W1,p(E) into Lq(E)Lq(E)
Topics on Weakly Differentiable Functions