Some properties of basic sequences in Banach spaces. (English) Zbl 0909.46007

Valdivia, Manuel

Rev. Mat. Univ. Complutense Madr. 10, No. 2, 331-361 (1997).

Summary: Some classes of basic sequences in Banach spaces are studied. We show in particular that if \(X\) is a Banach space with separable dual \(X^*\) and \(U\subset V\) are norming closed subspaces of \(X^*\), then there is a basic sequence \((x_n)\) in \(X\) such that, if \([x_n]\) is the closed linear hull of \((x_n)\) and \([x_n]^\perp\) is the subspace of \(X^*\) orthogonal to \([x_n]\), \(U+ [x_n]^\perp= V\) and the weak\(^*\)-closure of \(U\cap [x_n]^\perp\) in \(X^*\) coincides with \([x_n]^\perp\). This result, suggested by some problems in the quasi-reflexivity of Banach spaces, allows us to obtain some new results, as well as some already known ones, about this property. We also give here some results concerning Schauder basis in quotients of Banach spaces.

MSC:

46B15

Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

basic sequence; quasi-reflexivity; Schauder basis; quotients of Banach spaces