Functional analysis. Reprint of the 1966 original. (English) Zbl 0983.46001
Mineola, NY: Dover Publications. xii, 532 p. (2000).
The present book is a well organized introduction to functional analysis directed mainly to advanced undergraduate students of mathematics and may be used as an interesting addition by researchers in mathematical physics and engineering sciences. It consists of 29 chapters.
In chapters 1-10 the authors develop the necessary tools from linear algebra, metric spaces and topology, and introduce the reader to the main topic of this book: The Banach spaces and the Hilbert spaces.
Chapters 11-17 treat the celebrated Hahn-Banach theorem, the principle of uniform boundedness, and the closed graph theorem.
In chapters 18-23 are discussed spectral properties of linear operators and the reader can find here a condensed self-contained exposition of the Gelfand theory.
Chapters 23-28 offer several proofs of the spectral theorem for bounded, self-adjoint operators, and finally, the last chapter generalizes the preceeding results, removing boundedness from the hypothesis of the theorem.
[For a review of the 1966 original see Zbl 0141.11502].
In chapters 1-10 the authors develop the necessary tools from linear algebra, metric spaces and topology, and introduce the reader to the main topic of this book: The Banach spaces and the Hilbert spaces.
Chapters 11-17 treat the celebrated Hahn-Banach theorem, the principle of uniform boundedness, and the closed graph theorem.
In chapters 18-23 are discussed spectral properties of linear operators and the reader can find here a condensed self-contained exposition of the Gelfand theory.
Chapters 23-28 offer several proofs of the spectral theorem for bounded, self-adjoint operators, and finally, the last chapter generalizes the preceeding results, removing boundedness from the hypothesis of the theorem.
[For a review of the 1966 original see Zbl 0141.11502].
Reviewer: L.Janos (Kent/Ohio)
MSC:
| 46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |
| 47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |
| 47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |
| 46B10 | Duality and reflexivity in normed linear and Banach spaces |
| 46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
| 46A30 | Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness) |
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
