Marat V. Markin Bücher

The book presents a unique approach to measure and integration, beginning with a general abstract framework before focusing on Lebesgue measure and integration on the real line. It comprises nine chapters and an appendix, covering topics from basic set classes and measures to measurable functions and various convergence types. Key concepts include abstract Lebesgue integration, limit theorems, and comparisons between Lebesgue and Riemann integrals. Additionally, it explores Lp spaces, normed vector spaces, and signed measures. The text emphasizes a novel application of Lebesgue theory to enhance understanding of differentiation, broadening the classical total change formula to a wider range of functions. Designed for classroom use, it encourages active student engagement through 125 problems and 358 exercises, which require students to prove statements, fill in proof details, or provide counterexamples. More challenging problems are marked, and many include hints. The book is rich in examples and remarks that clarify definitions and provoke discussion on subtleties and essential conditions. This well-structured text is ideal for a one-semester graduate course in real analysis, appealing to students in mathematics, physics, computer science, and engineering, and preparing them for advanced studies in functional analysis and operator theory.

This book serves as an introductory text on functional analysis, designed for a one-semester graduate course within the standard U.S. curriculum. It comprises seven chapters and an appendix, guiding readers from the fundamentals of abstract spaces—such as metric, vector, normed vector, and inner product spaces—to the basics of linear operators and functionals. Key principles like the Hahn-Banach Theorem, Uniform Boundedness Principle, and Open Mapping Theorem are explored, along with their implications and applications, culminating in discussions on duality and reflexivity theory. The first chapter outlines essential preliminaries, while the appendix provides a concise overview of the Axiom of Choice and its equivalents. The text emphasizes active student engagement, featuring 112 problems essential for comprehension and progression. Many important statements are framed as problems, and 376 exercises throughout the text challenge students to prove or verify concepts. More complex problems are marked with an asterisk and often include hints. Rich in examples and remarks, the book addresses subtleties and questions regarding definitions and assertions. Prerequisites are intentionally low, making it accessible to a broad audience of STEM graduate students or advanced undergraduates with a solid calculus and linear algebra foundation. With ample applications and exercises, this text is ideal for students in mathematics, physic