Methods of Real Analysis

Methods of Real Analysis "After a typical calculus course in one real variable, you will find in this book an optimal, rigourous and clear introduction to real analysis. That means a closer inspection and generalization of old concepts (limit, convergence, continuiity, differentiaility, integrability... ) as well as encountering new mathematical objects and wider horizons in the same classic context). The work starts with sequences and series of real numbers. Then it says what a metric spaces M is and defines compactness, connectedness and completeness. as well as what means for a function to be continous at a given point of its domain. Later it reviews the Riemann integral and the basic properties of derivatives, offering a sound construction of the so called elementary functions. But is in the last four chapters 9-10-11-12 where the book show its virtues: Abel and Cèsaro summability, uniform convergence (but power series are missing!), uniform approximation (Stone-Weierstrass' and Ascoli-Arzela's theorem), Picard's existence theorem for ordinary differential equations, measure and Lebesgue integral on the real line (alas! without Lebesgue's ultimate version of the fundamental theorem of calculus, that express every absolutely continuous function as the integral of its -existing!- derivative, which is of course Lebesgue integrable) and finally a WONDERFUL (if too short) introduction to Fourier Analysis, that deals with Fourier series for both L_1 functions (the hard task) and L_2 functions (it includes Riesz-Fischer theorem). Note that for periodic functions L_2 is strctly smalller than L_1. Everything is done in a simple and direct style. I have decided to review this book because recently it has helped me a lot in understanding (for L_1 periodic functions) several tricky results. "Measure Theory and Fine Properties of Functions", now in a cheaper 2nd. edition or the classic General Theory of Functions and Integration, by A. E. Taylor. -- lim_bus"