Some New Classes in Topological Sequence Spaces Related to $L_r$-Spaces and an Inclusion Theorem for K(X)-Spaces

Abstract

The aim of the present paper is to get inclusion theorems for K(X)-spaces, that is, sequence spaces over any Frchet space X endowed with a K-topology (e.g. domains of operator valued matrices). Since Kalton’s closed graph theorem is an essential tool to get inclusion theorems in the case that Xequals the set of all complex numbers and since domains of operator valued matrices are not necessarily separable FK(X)-spaces we can no longer make use of FK-space theory. Therefore, it is necessary to develop new ideas to get inclusion theorems. For this we introduce two new classes of K(X)-spaces and prove a closed graph theorem for inclusion maps. One of them is closely related to the class of $L_r$-spaces introduced by Jinghui Qiu and to the closed graph theorem of J. Qiu, the other is connected with a well-known result of K. Zeller in summability theory. As an immediate corollary of the inclusion theorem proved in this paper we get a generalization of a theorem of Mazur-Orlicz type due to the authors.