Johnsonbaugh R., Pfaffenberger W.E. Foundations of Mathematical Analysis

New York: Dover Publ, Inc., 2010. — 411 p. — ISBN13: 978-0-486-47766-4.This book evolved from a one-year Advanced Calculus course that we have given during the last decade. Our audiences have included junior and senior majors and honors students, and, on occasion, gifted sophomores.
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 which are summarized in a brief appendix. Thus, theoretically, 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. Our intent is to teach students the tools of modern analysis as it relates to further study in mathematics, especially statistics, numerical analysis, differential equations, mathematical analysis, and functional analysis.Sets and Functions
The Real Number System
Set Equivalence
Sequences of Real Numbers
Infinite Series
Limits of Real-Valued Functions and Continuous Functions on the Real Line
Metric Spaces
Differential Calculus of the Real Line
The Riemann-Stieltjes Integral
Sequences and Series of Functions
Transcendental Functions
Inner Product Spaces and Fourier Series
Normed Linear Spaces and the Riesz Representation Theorem
The Lebesgue Integral