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A theorem on Schauder decompositions in Banach spaces

Publicacions de la Secció de Matemàtiques
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  • Mathematics


Pub . Mat . UAB Vol . 29 N4 1 Abril 1985 A THEOREM ON SCHAUDER DECOMPOSITIONS IN BANACH SPACES Abstract . In this paper we prove that in a Banach space all Schauder decompositions are slirinking iff all Schauder decompositions are boundedly complete . 1 . Definitions and preliminary results A sequence (x )m in a Banach space X is called a Schaudern n=1 basis if for every x E X there exists a unique sequence (an)n=1 in R such that x = anxn , and this series converges with respéct the n=1 norm of X . A sequence (yn)mn--1 is called a basic sequence if it is a basis of his closed linear span . tions Pn : X -. X defined by Miguel A . Ariño A Schauder decomposition of X is a sequence (X.)m of closedi i=1 subspaces of X such that for every x in X there exists a unique se-. quence (xi)i=1 mth x iE Xi for all i and x = ~ xi . Every Schauder i=1 decomposition of X is related with a sequence of continuous projec- Pn(x) -_ Pn( ~ xi ) _i=1 In all this paper, the linear span of an element x E X is denoted mby [x] and the closed linear span of the subspaces (Xi ) i=n (l< n <m< m) is denoted by [X . ] m - i i=n' The following theorem characterizes the Schauder decompositions and it can be found in [5] . 1 . Theorem : Let X be a Banach space and (Xn ) n=1 a sequence of closed- subspaces of X . The following are equivalent : i) (Xn)n=1 is a Schauder decomposition of X . ii) There exists a sequence (Pn)n=1 of continuous projections Pn : n such that Pn pm = Pmin(m n) and lim P (x)=x for., X -- " [Xi] i=1 every x in X . iii) There exists a sequence (Pn )ñ 1 of continuous projections Pn :- n ~such that Pn Pm = Pmin(m,n) and (Pn ) n=1 is uniformly-

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