Real Analysis and Foundations

Steven G. Krantz

• Verlag: Chapman and Hall/CRC
• Jahr: 2022
• ISBN:
  9781000593242
• Auflage: 5
• Thema: Mathematics
• Thema: Mathematical Analysis

Through four editions this popular textbook attracted a loyal readership and widespread use. Students find the book to be concise, accessible, and complete. Instructors find the book to be clear, authoritative, and dependable.

The primary goal of this new edition remains the same as in previous editions. It is to make real analysis relevant and accessible to a broad audience of students with diverse backgrounds while also maintaining the integrity of the course. This text aims to be the generational touchstone for the subject and the go-to text for developing young scientists.

This new edition continues the effort to make the book accessible to a broader audience. Many students who take a real analysis course do not have the ideal background. The new edition offers chapters on background material like set theory, logic, and methods of proof. The more advanced material in the book is made more apparent.

This new edition offers a new chapter on metric spaces and their applications. Metric spaces are important in many parts of the mathematical sciences, including data mining, web searching, and classification of images.

The author also revised the material on sequences and series adding examples and exercises that compare convergence tests and give additional tests.

The text includes rare topics such as wavelets and applications to differential equations. The level of difficulty moves slowly, becoming more sophisticated in later chapters. Students have commented on the progression as a favorite aspect of the textbook.

The author is perhaps the most prolific expositor of upper division mathematics. With over seventy books in print, thousands of students have been taught and learned from his books.

1 Real and Complex Numbers

DOI: 10.1201/9781003222682-2

1.1 The Real Numbers

This is a book about analysis in the real number system. Such a study must be founded on a careful consideration of what the real numbers are and how they are constructed. In this section, we give a careful treatment of the real number system. In the next, we consider the complex numbers.

We know from real numbers calculus that, for many purposes, the rational numbers are inadequate. It is important to work in a number system that is closed with respect to the operations we shall perform. This includes the limiting operations. While the rationals are closed under the usual arithmetic operations (addition, subtraction, multiplication, and division), they are not closed under the limits mathematical operation of taking limits. For instance, the sequence of rational numbers 3, 3.1, 3.14, 3.141, … consists of terms that seem to be getting closer and closer together, seem to tend to some limit, and yet there is no rational number that will serve as a limit (of course, it turns out that the limit is π—an “irrational” number).

We will now deal with the real number system, a system that contains all limits of sequences of rational numbers (as well as all limits of sequences of real numbers!). In fact, our plan will be as follows: in this section, we shall discuss all the requisite properties of the reals. The actual construction of the reals is rather subtle, and we shall put that in an Appendix to Section 1.1.

Definition 1.1: Let $A$ be an ordered set and $X$ a subset of $A$. The set $X$ is called bounded above if there is an element $b\in A$ such that $x\le b$ for all $x\in X$. We call...