FK-space

Abstract: Introduction. In [2] Bennett and Kahon proved the following property of the B K space ~ of bounded (real or complex) sequences: Given any separable FK--space F containing c o such that F n ~ is dense in ~ , o n e actually has ~ c F . The method of proof leading to this result is based on a detailed analysis of ~ and its subspaces using two-norm convergence. In the present paper we obtain the following generalization of the Bennett/Kalton result, using a different approach. We prove that for every BK-AB-space E having the so--called strong gliding humps property, the following is true: Given any separable FK-space F containing ~ such that F n E is dense in E , one has E c F. The method of proof we use to establish this result consists in checking the following two properties satisfied by every BK-AB-space E having the strong gl i ding humps property. Firstly, (i) every dense subspace D of E necessarily satisfies Dl3 = E 13 , and secondly, (ii) the topology o(EI3,E) is sequentially complete. Combining these facts yields the sequential completeness of o(Dl3,D) and therefore permits applying Kalton's closed graph theorem to the inclusion mapping (D,x(D,D~)) F, where F is a separable FK-space containing cI) such that D = F n E is dense in E. Both properties (i) and (ii) are of interest in themselves. Proving the sequential completeness of weak topologies of the form o(E~,E) involves techniques familiar in bounded consistency theory. We refer to [3] for a survey of these techniques. In fact, our present approach derives sequential completeness of the topology o(El3,E) from a weak form of the gliding humps property for the multiplier space M(E), closely related to the corresponding properties of M(E) considered in [3] and [8]. On the other hand, property (i) is related to the circle of problems connected with the