Mathematical analysis. The mastery of the implicitness (Q2883343)

The aim of the book under review is to impart a profound, versatile and stimulating introduction to several central topics of mathematical analysis, including the differential calculus of real functions, the fundamental properties of holomorphic functions of one complex variable, the related elements of general topology, the involved aspects of linear and polynomial algebra, the basic theory of real ordinary differential equations, a discussion of some numerical methods, and a description of asymptotic estimation techniques.NEWLINENEWLINE However, the book is certainly much more than just another of the thousands of analysis textbooks. In fact, as the author points out in the preface, this book is a product of his great fascination for the indirect method of cognition in mathematical research, especially for the huge variety of implicit objects, constructions, and results in mathematical analysis. It is both the great appeal and the stunning beauty of the subject that the author aspires to convey through this very remarkable, highly individual and special invitation to mathematical analysis and its various related areas in contemporary (French) university education.NEWLINENEWLINE As for the contents of the book, the material is arranged in seventeen chapters, each of which is divided into several sections. First, in a very prepossessing foreword, the author passionately explains both his motive and his conception for the current text. Immediately afterwards, concrete instructions concerning the various possible paths of working through the book are given, which certainly represents a useful service to the reader. Moreover, each chapter starts with an extra section titled ``Panorama'', thereby providing a further overview of the respective chapter, its particular aspects, and its methodological structure.NEWLINENEWLINE Chapter 1 discusses the metric topology of \(\mathbb{R}\) and \(\mathbb{R}^n\), together with covergent sequences and the concept of continuity for real functions on these spaces. Chapter 2 turns to extremal problems for continuous functions, their topological aspects, and some of their applications (both classical and more recent). The comparatively short Chapter 3 is devoted to the theorem of Rolle and Taylor's formula, while Chapter 4 deals with derivatives in dimension one, mean value theorems, and the basics of differential calculus in \(\mathbb{R}^n\) and in Banach spaces. Further elementary existence theorems appear in Chapter 5, chiefly the intermediate value theorem and a fixed-point theorem for continuous real functions of one variable.NEWLINENEWLINE Extremal problems in their differential aspects are analyzed in Chapter 6, with applications to real linear algebra, isoperimetric problems, the integral formula of Green-Riemann, trigonometric polynomials, and other contexts. Chapter 7 covers the basics of the theory holomorphic functions of one complex variable, where extremal properties of such functions serve again as the motivating theme. The concept of convexity and its significance in the study of extremal problems in real analysis are the topic of Chapter 8, including extrema of convex functions, extrema of convex sets, and the classical numerical methods of descent as particular algorithmic approaches. Chapter 9 explains contractions and fixed points of Lipschitz mappings, in varying generality, and gives the construction of fractal sets as a concrete application of the abstract results. This is followed by a comprehensive study of Brouwer's fixed-point theorem and some of its crucial applications in Chapter 10, with special emphasis on the different approaches (combinatorial, analytic, algebraic-topological, and functional-analytic) to the subject as well as an application to Nash equilibria in game theory. Chapter 11 offers a number of results illustrating the somewhat vague concept of ``regularity of solutions of implicit problems depending on a parameter'', and the subsequent Chapter 13 turns to two of the most fundamental results in mathematical analysis: the local inversion theorem and the implicit function theorem.NEWLINENEWLINE The determination of the roots of a (real or complex) polynomial, another field of action of implicit methods in mathematics, is the principal theme of Chapter 13. The central question is here how the roots of a polynomial vary with respect to modifications of the polynomial coefficients. Along the way, the reader encounters discriminants, symmetric functions, the orbit space \(\mathbb{C}^n\) modulo the symmetric group \({\mathcal S}^n\), the argument principle in complex analysis, and Rouch's theorem, among other related concepts and results. Chapter 14 is concerned with the analytic study of the eigenvalues of a (real or complex) matrix with the focus on symmetric matrices and the regularity of the mapping that associates the tuple of eigenvalues to a given matrix. In Chapter 15, another kind of (implicit) existence theorems is illuminated, namely the existence and local uniqueness of solutions of ordinary differential equations. Chapter 16 presents a number of numerical methods for constructing approximate solutions of polynomial equations, including the classical iterative techniques and some basic numerical linear algebra of matrices and their eigenvalues. Finally, Chapter 17 discusses several techniques of asymptotic estimation of implicitely defined objects in mathematics, ranging from reciprocal functions and implicitely defined sequences of numbers up to Newton polygons in the local study of plane algebraic curves.NEWLINENEWLINE Apart from the rich and versatile material developed in the course of the entire text, each chapter comes with an extra section of numerous, carefully selected and utmost instructive exercises. These exercises provide the reader with a wealth of additional, closely related and thematically enhancing topics, examples, methods, and results. Most of the exercises contain hints and indications for solutions, and each of the working problems appears in a multi-stage, systematically assembled and well-guided form, which helps the reader develop a solid ability for self-study, get acquainted with further aspects and theories, perceive the power of the methods and theorems presented in the main text, and see more of the fascinating beauty of mathematics and its coherency.NEWLINENEWLINE No doubt, this is a very special textbook on mathematical analysis, with a unique methodological and didactic flavor. It certainly reflects the author's passion for the subject just as much as his mastery of expository writing. In this respect, the book under review is a highly welcome (and remarkably inexpensive) addition to the vast textbook literature in the field.