Vilenkin N.Ya., Krein S.G. et al. Functional analysis

Groningen (Netherlands): Wolters-Noordhoff Publishing, 1972. — 394 p. OCRFunctional analysis originated at the beginning of the present century and became an independent mathematical discipline during the third and fourth decades; it developed rapidly and continues to do so. After the appearance of the remarkable book by the Polish mathematician S. Banach (Theorie des operations lineaires, Warsaw, 1932) the ideas and language of functional analysis permeated the most diverse branches of mathematics and its applications. This process has now gone so far that it is sometimes difficult to distinguish functional analysis from those fields in which it is applied.
On the other hand, the discussion of certain problems of classical functional analysis turned out to be rather restricted and this led to an examination of its basic concepts, that is, to a detailed analysis of its axiomatics. This process occurred during the past decade and can not yet be considered completed. We recall that I. M. Gelfand began his talk on functional analysis at the Fourth All-Union Mathematical Conference with the pessimistic words: "We still do not have a good definition of space, nor do we yet have a good definition of operator."
The authors of the present book were confronted by two dangers: to become lost in the numerous logical and conceptual sources of functional analysis or to become dissipated among the infinite number of branches in the delta of functional analysis as it flows into the sea of mathematical sciences. In order to avoid these dangers the autors strove to keep close to the main channel - the theory of operators and operator equations. The main material of this book is devoted to this theory. An exception is the extensive last chapter "Generalized functions" by N. Ya. Vilenkin which could also be part of a book on mathematical analysis since it contains the results of the influence of the ideas and methods of functional analysis on problems of mathematical analysis.