A counterexample to the approximation problem in Banach spaces

A Banach space B is said to have the approximation property (a.p. for short) if every compact operator from a Banaeh space into g can be approximated in the norm topology for operators by finite rank operators. The classical approximation problem is the question whether all Banach spaces have the a.p. In this paper we will give a negative answer to this question by constructing a Banach space which does not have the a.p. A Banach space is said to have the bounded approximation property (b.a.p. for short) if there is a net (Sn) of finite rank operators on B such that S,~I in strong operator topology and such that there is a uniform bound on the norms of the S~:s. It was proved by Grothendieek that the b.a.p, implies the a.p. and that for reflexive Banaeh spaces the b.a.p, is equivalent to the a.p. (see 1 p. 181 Cor. 2). So what we actually do in this paper is to construct a separable reflexive Banach space which fails to have a property somewhat weaker than the b.a.p.--the exact statement is given by Theorem 1. Since a Banaeh space with a Schauder basis has the b.a.p.