On a Generalization of the Köthe Coordinated Spaces

Abstract

The theory of sequence spaces introduced by Köthe and Toeplitz [11] and extensively developed by Köthe (see e.g. [12] or [13] for a full bibliography) has on the one hand given rise to theorems which have taken their place in the general theory of linear topological spaces and on the other hand led to generalizations of a more special nature in which the spaces considered and their duals consist of classes of integrable functions, for example the “espaces de Köthe” of Diexjdonné [6], the Banach function spaces studied by Luxemburg and Zaanen ([14] and particularly [15] for full bibliography) and by Ellis and Halperin [9]. The present author has studied a generalization of these spaces which takes as their characteristic feature the existence in them of a Boolean algebra of projectors and the fact that the elements of the dual are defined by the fact that they generate functions on this algebra continuous in topologies of a certain type. The other motivation of this theory is the theory of spectral multiplicity in Hilbert spaces [10], which suggests considering cases in which the topology on the Boolean algebra is generated by a noncountable family of measures. Such topologies are studied in [4] and the corresponding spaces in [5] for the case in which the algebras and the measures are countably complete and additive.

I wish to express my gratitude to the National Science Foundation of the U.S.A. whose grant GP-4070 in part supported the work for this paper and to the California Institute of Technology for their hospitality in allowing me the use of facilities for this work.