Exercises in analysis. II: Nonlinear analysis
When students today solve mathematical problems, they are not actually solving problems. Rather, they are doing exercises. The book under review trains readers to solve exercises in analysis while demonstrating the beauty and intricacy of this field of mathematics. This book is full of interesting problems. This is a useful collection of problems (with complete solutions) in pure and applied nonlinear functional analysis. The problems are selected so that working each one will teach you something; they are not drill, and do not yield to the obvious standard methods. The book covers a wide range of topics and would be suitable for supplementing an upper undergraduate course in analysis. The volume is also addressed to graduate students and to researchers in pure and applied analysis. The content of this book is divided into the following five chapters: 1. Function spaces; 2. Nonlinear and multivalued maps; 3. Smooth and nonsmooth calculus; 4. Degree theory and fixed point theory; 5. Variational and topological methods. At least 200 well selected problems are presented in the every of the first four chapters (176 exercises in the fifth chapter). The relevant main abstract results are summarized at the beginning of each chapter. The problems are challenging and accessible and they are ranging from easy to difficult, with level of difficulty designated by the authors. The higher problems are not more difficult, but require more advanced mathematics. Working through the problems will teach you a lot about the relevant results in pure and applied nonlinear functional analysis. The solutions run about a quarter of a page each (with a lot of variation), so they are not extremely difficult or especially easy, once you have hit upon the right facts to use. The book provides a good mix of abstract and concrete problems. The concrete problems are not simple drill and require some ingenuity and a good understanding of analytic concepts. Overall, this volume is an excellent gateway to the culture of problem solving. It is challenging and rewarding. The book shines a new light on mathematics and engages readers with its wonderful insights and problems.
scientific article; zbMATH DE number 6518674