A counterexample to the approximation problem in Banach spaces
A counterexample to the approximation problem in Banach spaces is a scholarly work, published in 1973 in ''Acta Mathematica''. The main subjects of the publication include approximation property, mathematics, pure mathematics, Banach space, fixed-point theorem, Eberlein–Šmulian theorem, discrete mathematics, and counterexample. 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 authors 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.