Zentralblatt MATH Database 1931 – 2010
c 2010 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
Zbl 1170.46002
MacCluer, Barbara D.
Elementary functional analysis. (English)
Graduate Texts in Mathematics 253. New York, NY: Springer. x, 207 p. EUR 34.95/net;
SFR 58.50; $ 49.95; £ 27.99 (2009). ISBN 978-0-387-85528-8/hbk; ISBN 978-0-38785529-5/ebook
http://dx.doi.org/10.1007/978-0-387-85529-5
This interesting book covers material from the basic elements of functional analysis
to more advanced topics such as spectral theory and is intended for specialists and
graduate students in mathematics and other disciplines. The prerequisites for reading
the book are general topology, linear algebra and real analysis (there is a short appendix
including measure theory and Lebesgue integral). Each chapter includes some historical
commentaries and many examples which make the book readable as well as a lot of
exercises which invite the reader on a long trip to the “functional analysis land”. The
book consists of the preface, six chapters, a bibliography containing 48 references and
subject index. The chapters of the book are:
(I) Hilbert Space Preliminaries and (II) Operator Theory Basics: These two chapters
provide the materials which are necessary for understanding the rest of book. The style
of the first two chapters is unique: A mixture of notions of Banach space and those
of Hilbert space. Dealing with Bergman spaces, the author gives a flavour of complex
analysis as well. The author uses the same notation for both the Hilbert adjoint and
the Banach adjoint. Some authors use A∗ for the first and A0 for the second.
(III) The Big Three: This chapter includes the following fundamental theorems: HahnBanach, Banach-Steinhaus, Open Mapping and Closed Graph. A story of Banach and
the Scottish Café is stated as well, see K. Ciesielski [Banach J. Math. Anal. 1, No. 1,
1–10 (2007; Zbl 1129.01007)].
(IV) Compact Operators: The author first examines norms on finite-dimensional spaces
and then introduces compact operators as the natural generalization of operators on
finite dimensional spaces. She also presents the spectral theorem for compact selfadjoint
operators.
(V) Banach and C ∗ -Algebras: This chapter is devoted to the continuous functional
calculus. A discussion of weak topologies without referring to locally convex topologies
is presented.
(VI) The Spectral Theorem: The main goal of this chapter is to explore the spectral
measure version of the spectral theorem for normal operators on a Hilbert space.
Mohammad Sal Moslehian (Mashhad)
Keywords : Banach space; Hilbert space; compact operator; normal operator; spectral theory; norm; weak topologies; Hahn-Banach theorem; Banach-Steinhaus theorem;
open mapping theorem; closed graph theorem; linear mapping; Lp -spaces; dual; Banach
algebra; C ∗ -algebra; spectral theorem; Gelfand transform
Classification :
∗ 46-01 Textbooks (functional analysis)
1
Zentralblatt MATH Database 1931 – 2010
c 2010 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
47-01 Textbooks (operator theory)
2