Topics in Linear and Nonlinear Functional Analysis

This textbook provides a concise introduction to linear and nonlinear functional analysis, preparing the reader for further study in areas such as spectral theory, partial differential equations, and probability theory. The first part develops classical linear theory. The opening three chapters follow in Hilbert's footsteps, culminating in the spectral theorem for compact operators. The fourth chapter introduces the fundamental results on Banach spaces, forming the core material for a first course in functional analysis. Three additional chapters treat more advanced topics, including convexity and weak topologies, the spectral theory of bounded operators, and operator semigroups. The second part is devoted to nonlinear functional analysis. Topics include calculus in Banach spaces, mapping degrees, and monotone maps. Throughout, the text adopts a student-friendly, bottom-up approach, beginning with concrete special cases and gradually progressing to more abstract results so that the underlying motivation remains clear. Numerous worked examples illustrate how the theory is applied in practice, and an extensive collection of exercises range from routine problems to more challenging ones. A solutions manual is available for instructors.