Mathematical Analysis

This book assumes a standard first course in analysis and gives a unified treatment of several topics taught in the last two years of an honours degree course. In the first chapter the standard topological notions are introduced in a metric space setting; complete metric spaces are defined and the contraction mapping theorem and Baire's theorem proved. The second chapter gives the standard properties of continuous functions between metric spaces; pointwise convergence of functions is studied and its shortcomings motivate a discussion of uniform convergence; among the theorems proved are Dini's, the Stone-Weierstrass and the Ascoli-Arzela. Chapter 3 gives further results on uniform convergence, in particular its relationship to the preservation of Riemann integrability and differentiability; the former, establishing the need for a more general integral, leads naturally to Chapter 4. Here the Daniel! extension procedure is applied to the lattice of continuous functions of compact support and the Riemann integral; the use of Dini's theorem makes this particularly neat. The relationship to the measure-theoretic approach is clearly explained. Finally double integrals are introduced and the Fubini and Tonelli theorems proved; here the extension procedure is applied to a lattice of step functions to avoid assuming knowledge of Riemann integration on the plane; in the reviewer's opinion it would have been preferable to take the continuous functions of compact support and the repeated integral, obtaining the equality of the two repeated integrals as an application of the Stone-Weierstrass theorem. The last chapter gives the L¹ and L² theories of the Fourier transform. This book is very attractive: the treatment is leisurely and while it is consistently rigorous many of the harder theorems have their proofs supplemented by informal explanations and graphical illustrations; also a number of interesting applications are given. Each chapter has a large number of interesting exercises some of which extend the theory while others give relevant counter-examples. Although the reluctance of undergraduates to buy books is to be deplored it must be reckoned with. The low price of this book, its suitability for several courses and the high standard of its presentation combine to make it ideal as a course book. FREDA E. ALEXANDER