Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0924.46002
Meise, Reinhold; Vogt, Dietmar
Introduction to functional analysis. Transl. from the German by M. S. Ramanujan. (English)
[B] Oxford Graduate Texts in Mathematics. 2. Oxford: Clarendon Press. x, 437 p. \sterling 47.50 (1997). ISBN 0-19-851485-9/hbk

For a review of the German original (1992) see 781.46001.\par 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.\par 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.\par 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.\par 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.\par 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.\par 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.
[P.Zabreiko (Minsk)]

MSC 2000:: *46-01 Textbooks (functional analysis)
47-01 Textbooks (operator theory)
46Bxx Normed linear spaces and Banach spaces
47A10 Spectrum and resolvent of linear operators
47A20 Extensions and related concepts of linear operators
47B25 Symmetric and selfadjoint operators (unbounded)
46A20 Duality theory of topological linear spaces

Keywords: Hahn-Banach theorem; duality theory; Fréchet spaces; Banach-Steinhaus theorem; normed linear spaces; Hilbert spaces; orthogonal systems; Fourier transformation; Sobolev spaces; compact operators in Banach and Hilbert spaces; Banach algebras; $C^*$-algebras; unbounded selfadjoint operators in Hilbert space; selfadjoint extensions; linear topological invariants; short exact sequences of Fréchet spaces; splittings

Citations: Zbl 0781.46001

Cited in: Zbl 1234.46001 Zbl 1219.46001 Zbl 1181.46001 Zbl 1133.46014 Zbl 1118.46005 Zbl 1085.46001 Zbl 1011.46003

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]