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Functional analysis with applications. (English) Zbl 0698.46001

New York etc.: John Wiley & Sons. xii, 344 p. £27.80 (1989).
Starting with metric spaces the authors have presented normed linear spaces, Hilbert spaces, Banach algebras, operator theory, fixed point theory, variational inequalities, complementary problems and sequence spaces. Important results such as the Baire category theorem, the Ascoli theorem, Open mapping theorem, closed graph theorem, the Riesz representation theorem, and the Gelfand-Mazur theorem are given with detailed proofs.
This is a text. So, there is some overlapping with earlier publications such as G. F. Simmons, Introduction to topology and modern analysis (1983; Zbl 0495.46001). On page 53, the first two elements of the sequence \(\{\zeta_ n\}\) are not specified. On page 6, analyticity of a function on a set need be mentioned. Some misprints are also there. A salient feature of the book is that it includes some solved problems. The following exercises typify the nature of the exercises:
1. Show that the closure of an ideal in a Banach algebra is an ideal.
2. Characterize the matrices \((\ell_{\infty},\hat c)\) and \((c,\hat c)\).
3. If T is a unitary operator on a Hilbert space, prove that \(\| T\| =1.\)
4. Let X be a metric space. If T is a contraction on X, then show that \(T^ n\) is a contraction for some n. If \(T^ n\) is a contraction on X for an \(n>1\), then show that T need not be a contraction.
Three Appendices, References, Index and List of Symbols are given. The get-up is satisfactory.

MSC:

46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B03 Isomorphic theory (including renorming) of Banach spaces
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
46A03 General theory of locally convex spaces

Citations:

Zbl 0495.46001
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