Schauder Decompositions and Completeness
نویسنده
چکیده
00 It x = S QnIf» m addition, the projections Pn = £ 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 (<2«)*=i and (Pn)"=i will always denote the corresponding sequences of projections as defined above.
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