Krantz Steven G. Real Analysis and Foundations

4th edition. — CRC Press, 2016. — 431 p.The new edition of this popular text is revised to meet the suggestions of users of the previous edition. A readable yet rigorous approach to an essential part of mathematical thinking, this text bridges the gap between classic theoretical texts and less rigorous ones, providing a smooth transition from logic and proofs to real analysis. Along with the basic material, the text covers Riemann-Stieltjes integrals, Fourier analysis, metric spaces and applications, and differential equations.The first three editions of this book have become well established, and have attracted a loyal readership. Students find the book to be concise, accessible, and complete. Instructors find the book to be clear, authoritative, and dependable.
We continue to listen to our readership and to revise the book to make it more meaningful for new generations of mathematics students.This new edition retains many of the basic features of the earlier versions.
We still cover sequences, series, functions, limits of sequences and series of functions, differentiation theory, and integration theory. The theory of functions of several variables is explored. And introductions to Fourier analysis and differential equations is included to make the book timely and relevant.In this new edition we endeavor to make the book accessible to a broader audience. We do not want this to be perceived as a “high level” text. Therefore we include more explanation, more elementary examples, and we stepladder the exercises. We update and clarify the figures. We make the sections more concise, and omit technical details which are not needed for a solid and basic understanding of the key ideas.The book assumes that the student has a solid background in calculus, and at least some experience with mathematical reasoning. We do not assume that the student knows topology or linear algebra. A typical student in a course using this book would be a junior with a major in an analytical science.
In the same spirit, we have eliminated Chapter 13 on advanced topics and Chapter 14 on normed linear spaces. These are very attractive sets of ideas, but are probably best treated in a more advanced course.We have updated and augmented the multivariable material in order to bring out the geometric nature of the topic. The figures are thus enhanced and fleshed out. Clearly functions of several variables is a suitable climax for this textbook.