Cambridge: Cambridge University Press (ISBN 0-521-84072-4/hbk; 0-521-60047-2/pbk). vii, 261 p. £ 24.99, $ 50.00/pbk; £ 50.00, $ 90.00/hbk (2004).
The book is designed for an advanced undergraduate two or three semester course of analysis. The topics of the course are presented relatively briefly, but correctly and rigorously. Some preliminary knowledge of calculus is expected, as well as the motivation of the students, as examples from “practical life” motivating particular mathematical constructions are quite rare. This enabled to cover a wide range of the subject, from the basic theory of real and complex numbers through functions of a real variable, metric spaces, Lebesgue measure and integration to advanced topics, like function spaces or Fourier series. Mathematical books usually respect the rule that no constructions should be used before they are rigorously explained. This supports the correctness of the text, but often results in an artificial ruptures in its inner logic. A typical example is the integral criterion for the convergence of infinite series. Usually the theory of infinite series preceeds the integrals, so the section on the integral criterion is often included into chapters dealing with integrals. The author breaks this rule by means of dividing the text into essential parts, which are self-contained, and advanced ones, denoted with an asterix, which expect some additional knowledge or present interesting non-trivial ideas. From this point of view the book opens new horizons in the second reading. Each section is divided into subsections and a set of exercises follows each subsection. The exercises are of both routine and theoretical nature.
The contents of the book is the following: 1. Introduction. Basic facts on real numbers and operations with them, construction of real numbers by Dedekind cuts. 2. The Real and Complex Numbers. Infimum and supremum, algebraic properties, decimal expansions, countability. An advanced part on algebraic and transcendental numbers. 3. Real and Complex Sequences. Basic properties of sequences and limits, extended real line. 4. Series. Convergence, absolute convergence, series with nonnegative terms, tests for convergence. An advanced part on conditional convergence, alternating series, including Riemann’s rearrangement theorem. 5. Power Series. Radius of convergence, differentiation of power series, operations with power series, Abel’s theorem. 6. Metric Spaces. Metric space, types of its subsets and points. Covering and compactness, Heine-Borel theorem, sequential compactness. An advanced part on the Cantor set. 7. Continuous Functions. Continuity of mappings between metric spaces, uniform continuity, properties of continuous functions on compact sets. The space of continuous functions on a real interval. Weierstrass polynomial approximation theorem. 8. Calculus. Differentiation of real- and complex-valued functions, mean value theorems, inverse functions. Riemann integral, its properties, Fundamental theorem of calculus. Taylor’s theorem. 9. Some Special Functions. Complex exponential function, Fundamental Theorem of algebra, infinite products, Euler’s formula for sine. 10. Lebesgue Measure on the Line. Introduction (including the Theorem of Banach and Tarski, proof is in the appendix at the end of the book). Outer measure, measurable sets and their properties. An example of a nonmeasurable set as an advanced section. 11. Lebesgue Integration on the Line. Measurable functions, Lebesgue integration, properties of the Lebesgue integral. Dominated convergence theorem, monotone convergence theorem, Fatou’s Lemma. 12. Integration and Function Spaces. Null sets, measurability almost everywhere. Connection between Riemann and Lebesgue integrals, Riemann integrability of real functions with null sets of discontinuity points. The spaces and . Differentiation of the integral, the Hardy-Littlewood inequality. 13. Fourier Series. Periodic functions, Fourier coefficients. The Bessel’s inequality. Dirichlet’s and Fejér’s theorems. The Weierstrass approximation theorem, the Riesz-Fischer’s theorem. An advanced part on the convolution. 14. Applications of Fourier Series. The Gibbs phenomenon. An example of a continuous, nowhere differentiable function. The isoperimetric inequality (among curves of a givel length the circle encloses the largest area). Weyl’s equidistribution theorem. Applications with partial differential equations. Fast Fourier transform. The Fourier integral. Uncertainity principle. The whole section is an advanced one. 15. Ordinary Differential Equations. Homogeneous linear equations, first order systems with constant coefficients. Existence and uniqueness, Peano theorem, Banach fixed point theorem, basic numerical methods.