This is a well written text on Real Analysis that may be used for a course in Advanced Calculus. It can also serve as a reference for advanced topics in Real Analysis.

The text begins with the elements of Set Theory, including cardinality. There is a detailed description of the construction of the Real Numbers as decimals. A very good feature here is the discussion of the arithmetic operations and the problem of computability with infinite decimals. The text contains very clear and detailed coverage of single and multivariable calculus. In addition, there are numerous examples of how the material is applied. The author includes examples and counterexamples that illustrate the material.

Some highlights of the text:

- Chapter 5 Functions: This chapter gives a detailed description of smooth functions, and how to construct them. There is material on real analytic functions and analytic continuation, something not always treated in texts designed for a first course in analysis. There is complete coverage of Fourier series. A nice part of this section is the discussion of the failure of uniform convergence in a Fourier series by considering the convergence properties of the Fourier series of the square wave function that introduces the Gibbs Phenomenon.
- Chapter 6: This chapter covers topics from Classical Analysis. These include the Gamma – Function and the Euler -Maclaurin Formula. There is a detailed section on Bernoulli numbers. These numbers are important in analysis and one surprising application of them is to evaluating sums of the form \( S_{r}(n)=1^{r} + 2^{r} + \ldots + n^{r} \) where \( n \) is a natural number and \( r \) is a nonnegative integer. This sum can be expressed as a finite sum involving binomial coefficients and Bernoulli Numbers. A complete discussion of this can be found in Emanuel Fischer's
*Intermediate Real Analysis*.
- Chapter 7: This chapter covers metric spaces. The coverage is detailed and very nearly a minicourse in metric spaces. It includes the Tietze Extension Theorem for continuous functions, the Cantor Set, completeness, and completion of a metric space. Continuing with the author’s desire to show the applicability of the material, there is a section on the Contraction Mapping Theorem and its use in initial value problems for ordinary differential equations.
- Chapter 8: This chapter covers fractals and iterated function systems, a topic not readily found in texts of this type.
- Chapter 9: The final chapter of the text covers Differential Calculus on \( \mathbb{R}^{m} \). The standard material is covered from the vector standpoint, including the concept of a normed vector space. The treatment is detailed and complete. Because of this chapter, it would be good if students have had an introductory course in Linear Algebra.

All in all *Essential Real Analysis* gives detailed coverage of topics in a first course in Real Analysis. The text covers a rigorous blend of classical and modern analysis. There may be more material than possible to cover in a semester, but there is much to choose from. This is a very well written text by an author who is well versed in the material.

Charles Traina, Ph.D., is a Professor of Mathematics at St. John’s University, Jamaica, N.Y. His research interests are in Group Theory and Measure Theory.