Geometric theory of Banach spaces. Part I. The theory of basis and minimal systems

In recent years substantial success has been achieved in the study of geometric and linear topological properties of Banach spaces ($B$-spaces). Our aim is to present some of the results that have been obtained since the appearance of the well-known survey of the geometric theory of $B$-spaces, the monograph of M. M. Day “Normed Linear Spaces”.
The term “geometric theory” which we use is to a large extent conventional. At present the principal method of investigating $B$-spaces is to study special sequences of elements of a space; this is more reminiscent of the methods of analysis than of geometry. In the first part of the survey we give an account of the apparatus of the theory of sequences and demonstrate its potential in investigating topological properties of Banach spaces. At the same time a general look at the whole host of facts, which make the current theory so rich, becomes possible in the study of the geometric structure of the unit sphere, that is, of the isometric properties of a space. This approach to the investigation of Banach spaces will be developed in a second part. These two parts do not exhaust the contemporary theory of normed linear spaces, which consists of at least two other large branches: the finite-dimensional Banach spaces or Minkowski spaces and the investigation of isomorphisms and embeddings. Each of these domains has recently received a fundamental stimulus to its development.
It is sufficient for the reader of the present article to be acquainted with the elements of functional analysis as given in Chapters 1–5 of [72] or in Chapters 1–4 of [45]. We shall omit the proofs of statements that are given in sufficient detail in the Russian literature, or that can be obtained by methods illustrated by other examples. In addition, we shall not mention proofs that would lead us away from the exposition of the method.