Kantorovitz S. Introduction to Modern Analysis

Oxford University Press 2003. — 447 p. — (Oxford Graduate Texts in Mathematics, 8)This book grew out of lectures given since 1964 at Yale University, the University of Illinois at Chicago, and Bar Ilan University. The material covers the usual topics of Measure Theory and Functional Analysis, with applications to Probability Theory and to the theory of linear partial differential equations. Some relatively advanced topics are included in each chapter (excluding the first two): the Riesz–Markov representation theorem and differentiability in Euclidean spaces (Chapter 3); Haar measure (Chapter 4); Marcinkiewicz’s interpolation theorem (Chapter 5); the Gelfand–Naimark–Segal representation theorem (Chapter 7); the Von Neumann double commutant theorem (Chapter 8); the spectral representation theorem for normal operators (Chapter 9); the extension theory for unbounded symmetric operators (Chapter 10); the Lyapounov Central Limit theorem and the Kolmogoroff ‘Three Series theorem’ (Application I); the Hormander–Malgrange theorem, fundamental solutions of linear partial differential
equations with variable coefficients, and Hormander’s theory of convolution operators, with an application to integration of pure imaginary order (Application II). Some important complementary material is included in the ‘Exercises’ sections, with detailed hints leading step-by-step to the wanted results. Solutions to the end of chapter exercises may be found on the companion website for this text: http://www.oup.co.uk/academic/companion/mathematics/kantorovitz.