This book aims to investigate how Riesz and Fredholm operator theories extend and adapt within the framework of topological vector spaces. At its core, this book seeks to generalize classical operator theory, traditionally developed in Banach and Hilbert spaces, to more inclusive topological settings. It offers a unified and comprehensive approach to understanding Riesz and Fredholm theories beyond the confines of local convexity and in the absence of an abundance of continuous linear functionals (which are the prevalent assumptions in existing literature). The exposition begins with foundational concepts in topology, including open and closed sets, neighborhoods, and compactness. It then introduces metrics, norms, and completeness, followed by a treatment of vector spaces, subspaces, and linear topological structures. The discussion progresses to inner products, orthogonality, and Hilbert space geometry. The heart of the book is the development of Riesz theory in topological vector spaces, beginning with compact operators, eigenvalues, and eigensubspaces, and advancing to spectral theory. This sets the stage for constructing a Fredholm theory in general topological contexts, requiring new definitions and a thorough exploration of the Fredholm index, its stability, and its applications. The book also includes a wide range of exercises (from foundational proofs to advanced problems) designed to reinforce the content covered and to provide additional insights.

Includes a wide range of exercises from foundational proofs to advanced problems Offers a unified and comprehensive approach to understanding Riesz and Fredholm theories Presents state-of-the-art treatment of Fredholm theory on topological vector spaces

Autorentext

Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Joel Shapiro are internationally recognized mathematicians with expertise spanning across the fields of Harmonic Analysis, Functional Analysis, and Partial Differential Equations. Dorina Mitrea, Irina Mitrea, and Marius Mitrea are Fellows of the American Mathematical Society and have collaborated extensively over the years. Their joint contributions include over a dozen of original research monographs, notably the comprehensive five-volume series Geometric Harmonic Analysis (Springer), The Hodge-Laplacian, now in its second edition (De Gruyter), Singular Integral Operators, Quantitative Flatness, and Boundary Problems (Birkhauser), Groupoid Metrization Theory (Birkhauser). Their work has significantly impacted harmonic analysis of geometric flavor. Joel Shapiro is a distinguished scholar renowned for his pioneering research in the topological and geometric aspects of functional analysis, particularly composition operators.

Inhalt

Topologic,Algebraic,andFunctionalAnalyticTools.- RieszTheoryinTopologicalVectorSpaces.- FredholmTheoryinTopologicalVectorSpaces.