zbMATH — the first resource for mathematics

An introduction to functional analysis. (English) Zbl 0751.46002
The book is devoted to the theory of linear spaces and linear operators between them. Here is the list of some of the distinguishing features of this book.
In Part I ( Topological vector spaces) the author deals not only with locally convex spaces. E.g. he presents Kakutani’s criterion of quasinormability of a topological vector space.
In Part II (The three basic principles) the uniform boundedness principle is proved in general matrix version. (Such versions were developed by P. Antosik, J. Mikusinski and their collaborators). This version is of interest because it can be used to prove the Nikodym boundedness theorem (and some other results).
In part III (Locally convex topological vector spaces) the author presents some results on quasibarrelled spaces and inductive limits (in addition to more standard topics).
Part IV (Linear operators) contains (besides standard material) such topics as: The space of Schwartz distributions, projecting the bounded operators onto the compact operators, weakly compact operators and absolutely summing operators.
Part V (Spectral theory) is devoted mainly to the spectral theory of operators in Hilbert spaces. In order to prove the spectral theorem for normal operators the author developed some theory of Banach algebras.
The book contains many examples and exercises.
On reviewer’s opinion this book will be useful for those who study or teach (linear) functional analysis.

46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
47A10 Spectrum, resolvent
46A03 General theory of locally convex spaces
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
46A08 Barrelled spaces, bornological spaces
46J05 General theory of commutative topological algebras
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)