In an elegant and concise fashion, this book presents the concepts of functional analysis required by students of mathematics and physics. It begins with the basics of normed linear spaces and quickly proceeds to concentrate on Hilbert spaces, specifically the spectral theorem for bounded as well as unbounded operators in separable Hilbert spaces. While the first two chapters are devoted to basic propositions concerning normed vector spaces and Hilbert spaces, the third chapter treats advanced topics which are perhaps not standard in a first course on functional analysis. It begins with the Gelfand theory of commutative Banach algebras, and proceeds to the Gelfand-Naimark theorem on commutative C*-algebras. A discussion of representations of C*-algebras follows, and the final section of this chapter is devoted to the Hahn-Hellinger classification of separable representations of commutative C*-algebras. After this detour into operator algebras, the fourth chapter reverts to more standard operator theory in Hilbert space, dwelling on topics such as the spectral theorem for normal operators, the polar decomposition theorem, and the Fredholm theory for compact operators. A brief introduction to the theory of unbounded operators on Hilbert space is given in the fifth and final chapter. There is a voluminous appendix whose purpose is to fill in possible gaps in the reader's background in various areas such as linear algebra, topology, set theory and measure theory. The book is interspersed with many exercises, and hints are provided for the solutions to the more challenging of these.
"A good text for an ambitious one-semester course or a more leisurely one-year course... The discussion is lively and clearly focused, with many illustrative examples. The author [blends] a large and varied selection of exercises into the text at the places where they can be most useful... The proofs are clear and thoughtfully organized with all the essential details provided."
"A fairly serious attempt has been made by the author at making the treatment almost self-contained...There is no doubt that this will become a standard reference on operator algebras and their spectral theory, written by one of the leading experts in the field."
Description du livre Birkhäuser, 1998. Hardcover. État : New. book. N° de réf. du libraire 3764358920