Klaus Deimling's "Nonlinear Functional Analysis" is a comprehensive survey of the main elementary ideas, concepts, and methods in nonlinear functional analysis. The book is written for graduate students and provides a thorough introduction to the subject, emphasizing both the formal aspects and the underlying spirit of the field. It begins with the topological degree in finite dimensions and progresses to infinite dimensions, covering topics such as Banach spaces, compact maps, set contractions, monotone and accretive operators, implicit functions, problems at resonance, multis, extremal problems, critical points of functionals, and bifurcation theory. The text is structured into ten chapters, each containing an introduction that explains the content and its relation to previous chapters. Each section ends with final remarks and exercises, ranging from simple to complex. The book uses an essentially uniform mathematical language and avoids overly complex algebraic or differential topological concepts, making it accessible to readers with a basic understanding of functional analysis. It includes numerous examples and applications, illustrating how abstract results can be applied to concrete models or other abstract contexts. The book also references a bibliography of relevant books, lecture notes, and survey articles, along with a list of selected research papers. The author thanks the publishers and colleagues for their contributions, highlighting the collaborative nature of the work. The text is organized to build upon previous knowledge, with each chapter and section logically leading to the next, providing a clear and structured approach to the subject of nonlinear functional analysis.Klaus Deimling's "Nonlinear Functional Analysis" is a comprehensive survey of the main elementary ideas, concepts, and methods in nonlinear functional analysis. The book is written for graduate students and provides a thorough introduction to the subject, emphasizing both the formal aspects and the underlying spirit of the field. It begins with the topological degree in finite dimensions and progresses to infinite dimensions, covering topics such as Banach spaces, compact maps, set contractions, monotone and accretive operators, implicit functions, problems at resonance, multis, extremal problems, critical points of functionals, and bifurcation theory. The text is structured into ten chapters, each containing an introduction that explains the content and its relation to previous chapters. Each section ends with final remarks and exercises, ranging from simple to complex. The book uses an essentially uniform mathematical language and avoids overly complex algebraic or differential topological concepts, making it accessible to readers with a basic understanding of functional analysis. It includes numerous examples and applications, illustrating how abstract results can be applied to concrete models or other abstract contexts. The book also references a bibliography of relevant books, lecture notes, and survey articles, along with a list of selected research papers. The author thanks the publishers and colleagues for their contributions, highlighting the collaborative nature of the work. The text is organized to build upon previous knowledge, with each chapter and section logically leading to the next, providing a clear and structured approach to the subject of nonlinear functional analysis.