A Note on Unconditional Bases

J. R. Retherford
1964 Proceedings of the American Mathematical Society  
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
more » ... 1.2) x=YlEi(x). i If the convergence in (1.2) is unconditional, we say that {R(E{)} is an unconditional basis of subspaces for X. If Ei is continuous for each i, we say the basis is a Schauder basis of subspaces. Indeed, if {x,} is a basis (of vectors) for X and {/¿} the sequence of biorthogonal coefficient functionals, then the one-dimensional subspaces spanned by x¿, i= 1, 2, ■ ■ ■ , form a basis of subspaces for X and the induced projections, Ei(x) =/¿(x)x¡, are necessarily continuous. We assume throughout this note that {Ei} is a sequence of continuous orthogonal projections and that X= [U^i R(E¡)], the linear closure of Ujl i R(E,). Thus all bases will be Schauder and in particular we will confine our attention to the unconditional Schauder bases of subspaces (u-bases) for X. If S denotes the collection of all finite subsets of the positive integers w, directed by inclusion, let i/o = \x: lim 2^ Ei(x) = x> , Vi= , v o-eS ¿eo-i \ o-eS »ei / = <x: lim 23 £<(*) = °°} and i/3 = X\tfi. It is clear that {R(E,)} is a u-basis if and only if X= i/o-It is also clear that U0C.U1, U2ÇZUz and U0 is everywhere dense in X.
doi:10.2307/2034905 fatcat:j4cxpd543jet5mdqpgmvsyxvra