Introduction to Modern Analysis
This book presents a nicely written exposition of important topics of measure theory, integration and functional analysis, with applications to probability theory and partial differential equations. The first two chapters deal with abstract Lebesgue integration theory (Lp-spaces, the Lebesgue-Radon-Nikodym theorem, complex measures, construction of measures and product measure). Chapter 3 deals with measure and topology. The Riesz-Markov representation theorem is proved and differentiation of measure in Euclidean spaces is discussed. Chapter 4 is devoted to conjugates of Lp-spaces as well as spaces of continuous functions and the Haar measure. Basic functional analysis is the object of chapter 5 (duality, Hahn-Banach theorems, weak topologies, the Krein-Milman theorem, the Stone-Weierstrass theorem and Marcinkiewicz’s interpolation theorem).
Chapter 6 is devoted to bounded operators (the uniform boundedness principle, the open mapping and closed graph theorems). Chapter 7 deals with Banach algebras and includes the Gelfand-Naimark-Segal representation theorem. Chapters 8-10 concentrate on Hilbert spaces, integral representation and unbounded operators. More advanced topics include the von Neumann double commutant theorem, spectral representation for normal operators and extension theory for unbounded symmetric operators. The chapter called ‘Application I’ deals with probability theory and ‘Application II’ deals with distributions and partial differential equations. The Hörmander-Malgrange theorem is proved and fundamental solutions of linear partial differential equations with variable coefficients are discussed. Some complementary material is included in the exercise sections. The book can be warmly recommended to university students and teachers of mathematics.