Some Aspects of Functional Analysis and Algebra

Functional analysis, which has become an independent branch of mathematics at the beginning of this century, occupies one of the central positions in contemporary mathematics. This is explained on the one hand by the fact that functional analysis made use of the main classical methods of analysis and algebra, and on the other hand by the rôle which functional analysis plays in contemporary physical science, especially in quantum physics. The study of mathematical problems connected with quantum mechanics was a turning point in the development of functional anlysis itself, and at the present time to a great extent it determines the main paths for its development. It can be said without exaggeration that contemporary functional analysis represents a new and serious step in the development of mathematics. In the last few years a number of new branches have arisen in functional analysis. Although relatively recently (about 20 to 30 years ago) functional analysis was thought of mainly as the theory of linear normed spaces, at the present time that theory, which is important and, roughly speaking, is completed, cannot even be considered as one of the basic branches of functional analysis. In general, functional analysis is still far from being completed, but the basic tendencies in its development are considerably clearer now than they were some 15 to 20 years ago. It is of course impossible to discuss in this paper all the basic questions of functional analysis. Therefore, we will limit ourselves to the consideration of a few selected problems. Although at a first glance the problems considered differ in character, common to all of them is, for instance, the fact that the development of each of these branches is closely connected with and is stimulated by