Functional analysis (basic ideas)
The paper is the survey of most important and perspective ideas of functional analysis in the present state and in the future. The brief history, basic ideas and perspectives are discussed in this last Kantorovich's paper. The most important classes of abstract spaces (Hilbert spaces, Banach spaces, metric linear spaces and locally convex topological spaces) are described and the main principles of classical functional analysis (Hahn- Banach principle of extension, Banach-Steinhaus principle of boundness and the open mapping or closed graph principle) are presented. The high attention is given to partially ordered vector spaces, explicitly to so- called K-spaces in which each bounded (in the set of order) set has a maximal element. The most important and promising field of application of functional analysis are, by the opinion of the author, the large economical problems. The paper contains a short exposition of applications of the theory of K-spaces to economical problems, namely to the problem of balances. The general formulation of fixed-point theorem for such spaces is given and the paper refers to some applications of this theorem to the problem of adequate construction of mathematical models in economy. One of the most perspective applications of functional analysis methods, from the point of view of the author, is the approximation of multidimensional large and incomprehensive models by simple ones in lower dimension saving the main characteristics of the problem.
scientific article; zbMATH DE number 4019721