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ON SOME PROBLEMS OF BANACH

The year 1972 marks the 80th birthday of Stefan Banach and the 40th anniversary of the appearance of his monograph [ 1 ]. In it he set out the main ideas and results of the theory of normed spaces, which are now classical. The book also contains many problems of varying degrees of difficulty. Some remain unsolved even today, although they have been the object of unrelenting attack by many mathematicians. Some of the problems are strictly for "sport", while others lie at the heart of functional analysis. In this lecture I wish to consider the following problems: 1) The approximation problem and the basis problem. 2) The problems of a metric and linear-topological characterization of Hilbert spaces in the class of all Banach spaces. 1. The approximation problem and the basis problem. In the study of properties of linear operators (operators, for short) it is convenient to be able to represent an operator as a limit of operators of simple structure. Since finite-rank operators are the best understood, it is reasonable to ask when a given operator can be represented as a limit of finite-rank operators. Since the limit in the operator norm of a sequence of finite-rank operators is compact, a natural question is: when does a given Banach space Υ have the following so-called approximation property. (A) Given any compact operator T: X -> Υ from some Banach space X to Y, there exist finite-rank operators Fn: X -> Υ such that