A structure theorem for echelon Köthe spaces

Abstract

In [9], Lopez-Molina defined the echelon Köthe spaces , which provide a suitable generalization of the echelon sequence spaces to a general measure space (see also [5]). In this paper, we show that the strutture of the separable echelon Köthe spaces is «nicely» close to the strutture of the echelon sequence spaces. Namely, our main result is: Let be a separable echelon Köthe space of order p, , with purely non-atomic. Then, there is an echelon sequence space so that is isomorphic to the space of all -summable sequences in . An an application we show that a separable echelon Köthe space has a basis (by a basis we mean a Schauder basis), which is unconditional if .