Schauder bases and Köthe sequence spaces

Abstract

In the light of the vast amount of literature on Schauder bases in Banach spaces which has appeared during the last thirty years, and considering the more recent development of the general theory of locally convex spaces, it is not surprising that many people are beginning to study Schauder bases in locally convex spaces. (See for example [1], [5], [7], [14], [16]-[21], [24].) One of the most important areas in the field of locally convex spaces is the theory of Kothe sequence spaces [10]. Not very much is gained, of course, by looking at the sequence space determined by a locally convex space with a Schauder basis. However, if one considers the case in which the topological dual and the Kothe dual coincide, then the two theories seem to come together very nicely. One obtains easy proofs of old theorems and important generalizations of others. Of course, as might be expected in any study involving Kothe sequence spaces, one obtains many counterexamples which mark out the boundaries of possible theorems. The authors feel that in this paper a new approach is introduced to both of the fields mentioned in the title and that although many new results are obtained, only the surface has been scratched. For example, no attempt has been made to study the specific types of locally convex spaces presently of interest to analysts, such as Frechet spaces, Q9,-spaces, Montel spaces, and nuclear spaces. It should be mentioned, however, that in almost all cases, the hypotheses of our general theorems are satisfied by these spaces. Aside from the overall approach, our main interest here is to study various types of Schauder bases which have previously been introduced, their relationships to each other and to various topological vector space concepts. In part I, we define the types of bases, interpret them in the context of a sequence space with a topology of uniform convergence on a family of subsets of the Kbthe dual and determine when a locally convex space can be viewed in this context. In part II we study the types of bases and their implications. The most important result is Theorem (1.10) which is a generalization of a theorem of R. C. James [9]. Finally in part III, we apply our results to the concepts of reflexivity and similar bases. Notation and terminology. If E[fT]; where E is a vector space over the field K of real or complex numbers and 9is a topology on E; is a locally convex space (always assumed to be Hausdorff) and (bn) is a sequence in E, we say that (bn) is a