Book Reviews

摘要

The book is an introduction to modern (Functional) Analysis for well-trained mathematical readers, with an Appendix discussing foundations and basic material in Probability Theory and Mathematical Statistics. The first two chapters cover basic material on Measure Theory, such as outer measures, extensions of measures on algebras, integration and L~p spaces, the Lebesgue-Radon-Nykodim theorem and issues related to convergence. The treatment is classical but highly readable, with a good number of exercises at the end of each chapter. The remaining part of the book goes deeper into the basic results of Functional Analysis, such as the Riesz Representation Theorem, the Hahn-Banach and Stone-Weierstrass Theorems, the analysis of Bounded Operators, Banach Algebras and Hilbert Spaces, the Spectral Theorem for Normal Operators, to conclude with the (technically more complicated) theory of unbounded operators. Among the exercises one can find several extremely interesting applications, in particular th trigonometric Hilbert basis of L~2 and Fourier series are dealt with in Chapter 8 (on Hilbert Spaces), and the Arzela-Ascoli Theorem on Chapter 9 (on Integral Representations).