A course in functional analysis. 2nd ed.

Abstract

The book under review is the second (updated) edition of thewell-known treatise on functional analysis for researchers and advancedstudents. [For a review on the first 1985-edition see Zbl 0558.46001.] \parIn comparison to the first edition, many new exercises and various commentshave been added, and the references have been updated. The main changerefers to the last chatper on Fredholm theory which has been completelyrevised and simplified. \parFor the reader's convenience, we brieflyrecall the headings of the eleven chapters of the book, which go as follows:I. Hilbert spaces. II. Operators on Hilbert space. III. Banach spaces. IV.Locally convex spaces. V. Weak topologies. VI. Linear operators on a Banachspace. VII. Banach algebras and spectral theory. VIII. C\sp*-algebras. IX.Normal operators on Hilbert space. X. Unbounded operators. XI. Fredholmtheory. \parIt seems worthwhile to emphasize those topics which aresomewhat beyond the scope of classical textbooks, namely the diagonalizationof selfadjoint compact operators with applications to Sturm-Liouvilleproblems in Chapter II, Banach limits and Runge's theorem on theapproximation of analytic functions by rational functions in Chapter III,and the fixed point principles of Schauder and Ryll-Nardzewski withapplications to Haar measures in Chapter V. At the end of the book, anAppendix is added on the dual spaces of the Lebesgue spaces L\sb p and thespace C\sb 0 of continuous functions vanishing at infinity. \parThe bookis a very readable and highly original contribution to the vast market oftextbooks on functional analysis. It should be valuable to a wide audienceof teachers and students.