Textbook in PDF format

This classroom-tested volume offers a definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. Upper-level undergraduate students with a background in calculus will benefit from its teachings, along with beginning graduate students seeking a firm grounding in modern analysis.
A self-contained text, it presents the necessary background on the limit concept, and the first seven chapters could constitute a one-semester introduction to limits. Subsequent chapters discuss 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, and the Lebesgue integral. Supplementary materials include an appendix on vector spaces and more than 750 exercises of varying degrees of difficulty. Hints and solutions to selected exercises, indicated by an asterisk, appear at the back of the book.
Our intent is to teach students the tools of modem analysis as it relates to further study in mathematics.
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