Barrelledness of Spaces with Toeplitz Decompositionsʹ

A Toeplitz decomposition of a locally convex space E into subspaces (E,) with projections (P,) is a decomposition of every x E E as x = C,P,x, where ordinary summability has been replaced by summability with respect to an infinite and lower triangular regular matrix. We extend to the setting of Toeplitz decompositions a couple of results about barrelledness of Schauder decompositions. The first result, given for Schauder decompositions by No11 and Stadler, links the barrelledness of a normed space E to the barrelledness of the pieces E, via the fact that Eʹ is big enough so as to coincide with its summability dual. Our second theorem, given for Schauder decompositions by Diaz and Miiiarro, links the quasibarrelledness of an No-quasibarrelled (in particular, (DF)) space E to the quasibarrelledness of the pieces Ek via the fact that the decomposition is simple