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Introduction to functional analysis. Transl. from the German by M. S. Ramanujan. (English) Zbl 0924.46002
Oxford Graduate Texts in Mathematics. 2. Oxford: Clarendon Press. x, 437 p. £ 47.50 (1997).
For a review of the German original (1992) see 781.46001. The first three parts of this book are an advanced course of functional analysis. Here one can find basic definitions and results of linear functional analysis. Part I: “Preliminaries” is introductory; it contains short descriptions of basic definitions and results form linear algebra, and the theory of metric and topological spaces. Part II: “Banach Spaces and Metric Linear Spaces” presents in detail the theory of normed linear spaces, the Hahn-Banach Theorem and Duality Theory, the Banach-Steinhaus Theorem and its consequences, the theory of Hilbert spaces and orthogonal systems, the theory of spaces $L_p(X,\mu)$ and ${\cal C}(X)'$, Fourier transformation and Sobolev spaces. Part III: “Spectral Theory of Linear Operators” is a good and exhausting account of basic results about linear operators. The theory of compact operators in Banach and Hilbert spaces, the general theory of Banach algebras and $C^*$-algebras, spectral theory of normal operators, the general and spectral theory of unbounded selfadjoint operators in Hilbert space, and even the theory of selfadjoint extensions of symmetric operators are accounted here. The last part IV: “Fréchet Spaces and Their Dual Spaces” is the most interesting in this book. The part has no analogues in other mathematical books and is a sufficiently comprehensive description of this field. Moreover, the authors present recent results on sequence spaces, linear topological invariants and short exact sequences of Fréchet spaces, and theorems about their splittings. Really, this part is a small research monograph. As one can see, the book is non-standard and interesting. Undoubtedly, it will be useful for all researchers and lecturers in the field. Of course, it will also be useful for graduate and post-graduate students studying functional analysis.

46-01Textbooks (functional analysis)
47-01Textbooks (operator theory)
46BxxNormed linear spaces and Banach spaces; Banach lattices
47A10Spectrum and resolvent of linear operators
47A20Dilations, extensions and compressions of linear operators
47B25Symmetric and selfadjoint operators (unbounded)
46A20Duality theory of topological linear spaces