This undergraduate textbook builds on Sally's Tools of the Trade: Introduction to Advanced Mathematics and requires a solid background in calculus and linear algebra. Two detailed appendices, covering nearly thirty pages, are references for the prerequisite number theory and linear algebra. This is a dense work, light on illustrations and with little in the way of flavoring asides. Exercises are sprinkled throughout the chapters, rather than gathered at the end. No solutions are provided, which is fine for a work that is not intended as a self-sufficient resource for independent readers. This also makes it an efficient reference or adjunct work to any assigned text. I would have been glad to have had it myself when I first encountered this material.
The handful of “Challenge Problems” leading off each chapter should serve as a focusing, if bracing, introduction to each chapter. Preceding this is often a paragraph-length introduction to the subject at hand. Typically, this is from a related important work by authors such as Banach and Toeplitz. Following are the expected definitions, theorems, and proofs seasoned with the aforementioned exercises. The chapters conclude with “Independent Projects.” These comprise aligned and select material that unifies and applies the theory. This supplemental content, designed to expand the reader's understanding, separates this text from others that may be considered. Examples of these projects include an exploration of spectral theory for compact self-adjoint operators on a Hilbert space and differentiability of a monotone function.
Sally begins by constructing the real and complex numbers, then explores metric theory, normed linear spaces, and differentiation in separate chapters. The chapter on integration follows Lebesgue, not Riemann, and the seven-chapter work concludes with Fourier analysis. Terms and notation are helpfully indexed separately. The elegant, complete, and rigorous presentation makes this an idea work for capable undergraduate and graduate students interested in learning and even teaching real analysis.
Tom Schulte teaches mathematics at Oakland Community College in Michigan.