Initiation à l'analyse appliquée / Jean-Pierre Aubin

This is a lively and stimulating introduction to a variety of topics in applied analysis. The expert author points out well that a certain level of abstraction allows to use mathematical analysis with advantage in quite different fields of applications, including pattern recognition, neural networks, noncooperative games, market equilibria, dynamical systems, optimization, and so on. A main didactic idea is to handle the principal topological concepts by confining to metric spaces. The reader should be familiar with the basic facts of calculus and linear algebra in finite dimensions; the other mathematical prerequisites are outlined in the text. The book starts by an excellent “Epigraph” and a helpful “Reader's guide to choose his/her route”. Section 1 is devoted to examples of metric spaces and to Hölder and Minkowski's inequalities. In Section 2, the construction principles of metric spaces are presented, including the notions of metrizable and uniform spaces, Fréchet spaces, uniform convergence. Section 3 and Section 4 broadly handle topological properties of metric spaces and the standard subjects on continuity. Section 5, “Optimization”, includes, e.g., optimization and variational principles, upper/lower limits, convex functions, mathematical morphology, fuzzy sets, Ky Fan's inequality, applications in economics. Section 6, “Pseudo-inverses and nonlinear equations”, contains, e.g., applications of pseudo-inverses, tensor products, fixed point theorems, differentiable maps, implicit function theorem, nonlinear equations. Section 7 is concerned with basic notions and results in set-valued analysis. Finally, Section 8 presents an interesting selection of topics related to differential equations, e.g., Cauchy-Lipschitz, Peano and Nagumo's Theorems, replicator systems, invariance and viability, Lyapunov functions, first-order partial differential equations. (Zentralblatt)

Bibliogr. p. [385]-389. Index