Type: Article
Publication Date: 1973-09-01
Citations: 44
DOI: https://doi.org/10.2140/pjm.1973.48.113
Results of H. P. Rosenthal and the author on w*-basic sequences are combined with known techniques and applied to quasi-complementation problems in Banach spaces.l Introduction* Recall that (closed, linear) subspaces Y, Z of the Banach space X are quasi-complements (respectively complements)Suppose that Y, Z are quasi-complements, but not complements, for the separable space X.We show that there exist closed subspacesThis generalizes a theorem of James [5], who proved the existence of Yi for the case of general separable X and the existence of Y 2 for separable, reflexive X.Our proof uses James' method (and w*basic sequences), but seems simpler than James' construction.Also, our argument provides information for some nonseparable spaces.We show also the following.THEOREM 2. Suppose Y is a subspace of X and F* is weak*separable.If X/Y has a separable, infinite dimensional quotient space, then Y is quasi-complemented in X.