Analysis Now [Paperback]

Graduate students in mathematics, who want to travel light, will find this book invaluable; impatient young researchers in other fields will enjoy it as an instant reference to the highlights of modern analysis. Starting with general topology, it moves on to normed and seminormed linear spaces. From there it gives an introduction to the general theory of operators on Hilbert space, followed by a detailed exposition of the various forms the spectral theorem may take; from Gelfand theory, via spectral measures, to maximal commutative von Neumann algebras. The book concludes with two supplementary chapters: a concise account of unbounded operators and their spectral theory, and a complete course in measure and integration theory from an advanced point of view.Graduate students in mathematics, who want to travel light, will find this book invaluable; impatient young researchers in other fields will enjoy it as an instant reference to the highlights of modern analysis. Starting with general topology, it moves on to normed and seminormed linear spaces. From there it gives an introduction to the general theory of operators on Hilbert space, followed by a detailed exposition of the various forms the spectral theorem may take; from Gelfand theory, via spectral measures, to maximal commutative von Neumann algebras. The book concludes with two supplementary chapters: a concise account of unbounded operators and their spectral theory, and a complete course in measure and integration theory from an advanced point of view.1 General Topology.- 1.1. Ordered Sets.- The axiom of choice, Zorns lemma, and Cantorss well-ordering principle; and their equivalence. Exercises..- 1.2. Topology.- Open and closed sets. Interior points and boundary. Basis and subbasis for a topology. Countability axioms. Exercises..- 1.3. Convergence.- Nets and subnets. Convergence of nets. Accumulation points. Universal nets. Exercises..- 1.4. Continuity.- Continuous functions. Open maps and homeomorphisms. l��