RMC - Vol. 10, Num. 2 - Manuel Valdivia

Full paper in PDF:
$%M. Valdivia, Some properties of basic sequences in Banach spaces, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 331–361.%$

Some Properties of Basic Sequences in Banach Spaces

Manuel VALDIVIA

Departamento de Análisis Matemático
Universidad de Valencia
Dr. Moliner 50
46100 Burjasot (Valencia) — Spain

Received: January 8, 1997
Revised: April 28, 1997

ABSTRACT

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 $U < V$ are norming closed subspaces of $* X$ , then there is a basic sequence $(xn)$ in $X$ such that, if $[xn]$ is the closed linear hull of $(xn)$ and $_L [xn]$ is the subspace of $* X$ orthogonal to $[xn]$ , $_L U + [xn] = V$ and the weak^*-closure of $_L U /~\ [xn]$ in $* X$ coincides with $_L [xn]$ . 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.

1991 Mathematics Subject Classification: 46B15.