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**A course in functional analysis.
2nd ed.**
*(English)*
Zbl 0706.46003

Graduate Texts in Mathematics, 96. New York etc.: Springer-Verlag. xvi, 399 p. DM 148.00 (1990).

The book under review is the second (updated) edition of the well-known treatise on functional analysis for researchers and advanced students. [For a review on the first 1985-edition see Zbl 0558.46001.]

In comparison to the first edition, many new exercises and various comments have been added, and the references have been updated. The main change refers to the last chatper on Fredholm theory which has been completely revised and simplified.

For the reader’s convenience, we briefly recall 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 Banach space. VII. Banach algebras and spectral theory. VIII. \(C^*\)-algebras. IX. Normal operators on Hilbert space. X. Unbounded operators. XI. Fredholm theory.

It seems worthwhile to emphasize those topics which are somewhat beyond the scope of classical textbooks, namely the diagonalization of selfadjoint compact operators with applications to Sturm-Liouville problems in Chapter II, Banach limits and Runge’s theorem on the approximation of analytic functions by rational functions in Chapter III, and the fixed point principles of Schauder and Ryll-Nardzewski with applications to Haar measures in Chapter V. At the end of the book, an Appendix is added on the dual spaces of the Lebesgue spaces \(L_ p\) and the space \(C_ 0\) of continuous functions vanishing at infinity.

The book is a very readable and highly original contribution to the vast market of textbooks on functional analysis. It should be valuable to a wide audience of teachers and students.

In comparison to the first edition, many new exercises and various comments have been added, and the references have been updated. The main change refers to the last chatper on Fredholm theory which has been completely revised and simplified.

For the reader’s convenience, we briefly recall 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 Banach space. VII. Banach algebras and spectral theory. VIII. \(C^*\)-algebras. IX. Normal operators on Hilbert space. X. Unbounded operators. XI. Fredholm theory.

It seems worthwhile to emphasize those topics which are somewhat beyond the scope of classical textbooks, namely the diagonalization of selfadjoint compact operators with applications to Sturm-Liouville problems in Chapter II, Banach limits and Runge’s theorem on the approximation of analytic functions by rational functions in Chapter III, and the fixed point principles of Schauder and Ryll-Nardzewski with applications to Haar measures in Chapter V. At the end of the book, an Appendix is added on the dual spaces of the Lebesgue spaces \(L_ p\) and the space \(C_ 0\) of continuous functions vanishing at infinity.

The book is a very readable and highly original contribution to the vast market of textbooks on functional analysis. It should be valuable to a wide audience of teachers and students.

Reviewer: P.P.Zabreiko

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47A53 | (Semi-) Fredholm operators; index theories |