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A Note on Unconditional Bases

1964
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Proceedings of the American Mathematical Society
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1. Definitions and remarks. A sequence of nontrivial subspaces {M i} of a Banach space X is a basis of subspaces for X if and only if for every x(EX, x can be uniquely written 00 (1.1) x = 23 Xi, Xi £ Mi. i If {Mi} is a basis of subspaces for X, then the linear operators defined by Ei(x)=Xi, where x= 23™ xit x¿£Af" form a sequence of orthogonal projections (E{ = E2; £¿£y = 0, i^j). If i?(£¿) denotes the range of £,-, then clearly i?(£¿) = M, and each x£X can be expressed 00 (1.2) x=YlEi(x). i

doi:10.2307/2034905
fatcat:j4cxpd543jet5mdqpgmvsyxvra