Real analysis
"Real Analysis builds the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in R[superscript n]. It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets. It gives the three characterizations of compactness: as "closed and bounded," via sequences, and via open covers. Topics include Fourier series, the Gamma function, metric spaces, and Ascoli's Theorem." "The text not only provides efficient proofs, but also shows the student how to come up with them. The exercises come with select solutions in the back. Here is a real analysis text that is short enough for the student to read and understand and complete enough to be the primary text for a serious undergraduate course."--Jacket
Print Book, English, ©2005
American Mathematical Society, Providence, R.I., ©2005
viii, 151 pages : illustrations ; 26 cm
9780821836705, 0821836706
58451888
Part I: Real numbers and limits: Numbers and logic Infinity Sequences Functions and limits Part II: Topology: Open and closed sets Continuous functions Composition of functions Subsequences Compactness Existence of maximum Uniform continuity Connected sets and the intermediate value theorem The Cantor set and fractals Part III: Calculus: The derivative and the mean value theorem The Riemann integral The fundamental theorem of calculus Sequences of functions The Lebesgue theory Infinite series $\sum a_n$ Absolute convergence Power series Fourier series Strings and springs Convergence of Fourier series The exponential function Volumes of $n$-balls and the gamma function Part IV: Metric spaces: Metric spaces Analysis on metric spaces Compactness in metric spaces Ascoli's theorem Partial solutions to exercises Greek letters Index.
Includes index