This is an excellent book, well worth considering for a textbook for an undergraduate analysis course. The approach encourages global thinking by introducing and using normed vector spaces and Banach spaces from the beginning. The writing is first-rate and the book has been prepared with great care; I found very few misprints.
This may seem self-serving, but I really think there's a better approach to Riemann-Stieltjes integrals than the standard one presented in most books. See Chapter 7 of the book under review and compare with Section 35 in my book, Elementary Analysis: The Theory of Calculus. My approach avoids anomalies in the standard approach without losing anything useful.
For example, Theorem 7.2.4 tells us that if an integrand and integrator are both discontinuous at the same point, then the integrand is not Riemann-Stieltjes integrable. This and other anomalies disappear using my definitions, which are a bit more intricate than the standard ones. The key improvement is that jump functions are now integrators in the natural manner.
Kenneth A. Ross (firstname.lastname@example.org) taught at the University of Oregon from 1965 to 2000. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor.
PART I. ADVANCED CALCULUS IN ONE VARIABLE.
1. Real Numbers and Limits of Sequences.
2. Continuous Functions.
3. Rieman Integral.
4. The Derivative.
5. Infinite Series.
PART II. ADVANCED TOPICS IN ONE VARIABLE.
6. Fourier Series.
7. The Riemann-Stieltjes Integral.
PART III. ADVANCED CALCULUS IN SEVERAL VARIABLES.
8. Euclidean Space.
9. Continuous Functions on Euclidean Space.
10. The Derivative in Euclidean Space.
11. Riemann Integration in Euclidean Space.
Appendix A. Set Theory.