Zeitschrift für Analysis und ihre Anwendungen
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Published online: 1993-03-31
Some New Classes in Topological Sequence Spaces Related to $L_r$-Spaces and an Inclusion Theorem for K(X)-SpacesJohann Boos and T. Leiger (1) Fernuniversität-GHS, Hagen, Germany
(2) Tartu University, Estonia
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.
Keywords: Topological sequence spaces (over an F-space), summability in abstract structures, (operator valued) matrix maps, inclusion theorems, consistency of (operator valued) matrix maps, closed graph theorems
Boos Johann, Leiger T.: Some New Classes in Topological Sequence Spaces Related to $L_r$-Spaces and an Inclusion Theorem for K(X)-Spaces. Z. Anal. Anwend. 12 (1993), 13-26. doi: 10.4171/ZAA/582