Schauder Decompositions and Completeness

00 It x = S Qn x - If» m addition, the projections P n = £ Q,- are equicontinuous, then n = 1 «= 1 (£n)^°=1 is said to be an equi-Schauder decomposition of E. It is obvious that a Schauder basis is equivalent to a Schauder decomposition in which each subspace is one-dimensional, and that it is equi-Schauder if and only if the corresponding decomposition is equi-Schauder. For more information on Schauder decompositions see, for example [2 and 3]. In this paper, it will be shown that if E is locally convex and possesses an equiSchauder decomposition, the properties of sequential completeness, quasicompleteness or completeness of E may be related very simply to the properties of the decomposition; and that if £ possesses an equi-Schauder basis, these three types of completeness are equivalent. If (£„)*=! is a Schauder decomposition of E, the sequences ( polar topology on E. Suppose (En)™=1 is an equi-Schauder decomposition for (E, T) and let (xa)a eAbe a x-Cauchy net on E such that for each n (Qn xa)a e A converges. Then: (i) (lim Pn xa)"-i ' s a ?-Cauchy sequence. a