Foundations of Mathematical Analysis covers real analysis—from naïve set theory and the axioms for the real numbers to the Lebesgue integral, with sequences and series, metric spaces, the Riemann-Stieltjes integral, inner product spaces, Fourier series, Tauberian theorems, the Riesz representation theorem, and a brief discussion of Hilbert spaces in between.

The book evolved from a one-year Advanced Calculus course that both authors taught during the late 1960s and throughout the 1970s. The audience included junior and senior majors and honors students, and, on occasion, gifted sophomores. Professor Pfaffenberger has continued to teach this course. (Professor Johnsonbaugh moved to computer science.) The intervening years have confirmed the importance of real analysis. Analysis is a core subject in mathematics and is a prerequisite for further study in mathematics. Analysis is also fundamental to many related fields such as statistics. Several of Professor Pfaffenberger's students have completed doctorates at distinguished institutions (e.g., Princeton, Harvard, Berkeley, Cambridge), and many have specialized in analysis.

Because we believe that an essential part of learning mathematics is doing mathematics, we have included over 750 exercises, some containing several parts, of varying degrees of difficulty. Hints and solutions to selected exercises, indicated by an asterisk, are given in the back of the book.

The material is logically self-contained; that is, all of our results are proved and are ultimately based on the axioms for the real numbers. We do not use results from other sources except for a few results from linear algebra that are summarized in a brief appendix. Thus, logically, no prerequisites are necessary to understand this material. Realistically, the prerequisite is some mathematical maturity such as one might acquire by taking calculus and, perhaps, linear algebra.

It is our belief that understanding the limit concept is the key to a sound foundation for the study of analysis. Thus, after initial chapters on sets and the real number system (Chapters I–III), we introduce the limit concept using numerical sequences and series (Chapters IV and V). Chapter VI then covers the limit of a function. Theorem 16.6 in Chapter IV contains an interesting proof that (1+1/n)ⁿ is increasing and convergent. A discussion of double series (Section 29) leads to a quick proof that a rearrangement of an absolutely convergent series converges and the sum of the rearranged series is equal to the sum of the original series (Theorem 29.7). Double series are also used to discuss the Cauchy product of two series (Theorem 29.9) and to give some results on power series (Corollary 29.10 and Theorem 29.11).

In Chapter VII, we move to the general setting of metric spaces. Results include the Bolzano-Weierstrass characterization of a compact metric space (Section 43) and the Baire category theorem (Section 47). The first seven chapters could be used for a one-term course on the Concept of Limit.

After a review of differential calculus (Chapter VIII), Chapter IX gives a detailed introduction to the theory of Riemann-Stieltjes integration. We discuss measure zero (Section 57) and give a necessary and sufficient condition for the existence of the Riemann integral (Section 58).

We then turn to the study of sequences and series of functions (Chapter X). We provide applications to power series (Section 63) and Abel's limit theorems (Section 64). We also discuss summability methods and Tauberian theorems (Section 65).

Chapter XI discusses the exponential, logarithm, and trigonometric functions. The exponential, sine, and cosine functions are defined by power series, after which their standard properties are derived. The other trigonometric functions are defined in terms of the sine and cosine functions, and the logarithm function is defined as the inverse of the exponential function.

Inner product spaces and Fourier series are the topics for Chapter XII. Included is a discussion of Cesàro summability (Section 77) and Hardy's Tauberian theorem (Theorem 79.1) with an application to Fourier series (Theorem 79.3).

Chapter XIII develops normed linear spaces and proves a version of the Riesz representation theorem (Theorem 84.1).

The last chapter (Chapter XIV) studies the Lebesgue integral. Topics include measurable functions (Section 87), integration on positive measure spaces (Section 88), and Lebesgue measure (Sections 89 and 90). The concluding section (Section 91) introduces Hilbert spaces and proves the Riesz-Fischer theorem (Theorem 91.9).

The book concludes with an appendix on vector spaces, a list of references, and hints to selected exercises. The appendix summarizes some of the key definitions and theorems concerning vector spaces that are used in the book.

The book contains nearly 100 worked examples. These examples clarify the theory, show students how to develop proofs, demonstrate applications of the theory, elucidate proofs, and help motivate the material.