On certain echelon spaces (Q1071213)

The echelon spaces of order (p,q) \((1<p,q<\infty)\), are reflexive. In general, the echelon spaces of order (1,p) or (p,1) are not reflexive. It is known that the echelon spaces of order (1,1) (the Köthe echelon spaces when its elements are considered as double sequences) are reflexive if and only if they are Montel spaces. But the echelon spaces of order (1,p) or (p,1) \((1<p<\infty)\) do not verify this condition. In this paper a precise criterion is given for this spaces to be reflexive spaces. It is proved also that an echelon space of order (p,q) is not Schwartz if and only if it has a quotient topologically isomorphic to \(\ell^ p\) or \(\ell^ q\).