On Köthe spaces

Abstract

1. Kothe and Toeplitz in [7] and later Kothe in [8] and [9] began the study of certain pairs of subspaces of the space of all real sequences. Their theory has been generalized by Dieudonne [4], M. J. Cooper [3], and Lorentz and Wertheim [11]. Dieudonne, in his work considers pairs (A, A*) of subspaces of the space of locally integrable functions on a locally compact (-compact Hausdorff space E with a Radon measure ,u. The spaces (A, A*), which are locally convex topologically complete vector lattices, are called K6the spaces; well-known examples of these are the pairs (Lp, Lq). We present two results which contribute to the theory as developed by Dieudonne; however, since the topology of the underlying topological space E plays no role in these results, they are presented in a purely measure theoretic setting. This necessitates presenting some of the general theory of K6the spaces. The first of the two results generalizes the classical theorem: if Lrc LS and r # s, then L' is of the first category in Ls; the second result proves the equivalence of four statements; one of which is: A* can be identified with the topological dual of A.