An introduction to Hilbert space. (English) Zbl 0645.46024

Cambridge Mathematical Textbooks. Cambridge (UK) etc.: Cambridge University Press. 239 p. £9.95/pbk (1988).
The present book is intended primarily for an undergraduate audience. The authors believes that a sound grounding in Hilbert space theory is the best way how to approach functional analysis. It consists of sixteen chapters dealing with the following topics: Inner product spaces, Normed spaces, Hilbert and Banach spaces, Orthogonal expansions, Classical Fourier series, Dual spaces, Linear operators, Compact operators, Sturm- Liouville systems, Green’s functions, Eigenfunction expansions, Positive operators and contractions, Hardy spaces, Approximation by analytic functions and approximation by meromorphic functions. This last chapter and the one concerning the positive operators may be of interest to electrical engineers, since some recent developments, particularly in control and filter design, require familiarity with this aspect of operator theory. The book presupposes introductory courses in real analysis, linear algebra, topology of metric spaces and elementary complex analysis. The chapter concerning Hardy spaces requires a certain familiarity with Lebesgue measure.
Reviewer: L.Janos


46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46C99 Inner product spaces and their generalizations, Hilbert spaces
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
34L99 Ordinary differential operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
41A30 Approximation by other special function classes
47A10 Spectrum, resolvent
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
46B03 Isomorphic theory (including renorming) of Banach spaces