Concept Of Vector Space Research Articles

Schauder bases and Köthe sequence spaces

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

Abstract vector spaces applied to problems in detection and estimation theory

The concepts of abstract vector spaces are summarized and applied to such problems as the detection of binary signals in additive Gaussian noise and the maximum a posteriori estimation of analog signals transmitted over random fading media. Although the results obtained are in most cases well known in the literature, it is felt that the abstract vector approach provides notational and manipulative simplicity that warrants its extension to other problems in detection and estimation theory.

System theory

I have attempted to call attention to what is perhaps the weakest link in system theory, namely, the process of making measurements which yield the numbers that serve as the inputs to the mathematical theory. I have called attention to Dirac's notation because quantum mechanics, in which this notation was first used, is very much concerned with the measurement process and the notions of linear operators which are used therein have a very close tie to measurement problems in larger systems. Secondly, I have suggested that we need more invention and critical innovation in developing a notation well suite(d for the representation and study of systems. Here, vector-space concepts and matrix notation may be used to good advantage, and signal-flow graphs provide a new kind of algebraic notation that is well matched to the visual sense and deserves continued study. Finally, the links between the theory and the natural universe may be strengthened by extending the rich stockpile of abstract concepts downward to acquire ever fuller itnterpretations and insights into physical phenomena. Let us not forget Poincare's answer to the question, "Who has taught us the true analogies, the profound analogies which the eyes do not see, but which reason can divine? It is the mathematical mind, which scorns content and clings to pure form."