摘要
本文部分回答了R.Holub提出的关于基的Hahn-Banach延拓的两个问题。证明了如果{x_n}_(n=1)~∞是X的基序列,使得[x_n]_(n=1)~∞在X中可补,则存在X上的一个等价范数‖.‖,使得{x_n}_(n=1)~∞的系数泛函{x_n}关于这个等价范数‖.‖具有一个Hahn-Banach延拓{f_n}_(n=1)~∞,且{f_n}_(n=1)~∞仍然是基序列。我们还证明如果{x_n}_(n=1)~∞是X的一个基序列,使得[x_n]_(n=1)~∞在X中可补,且{x_n}_(n=1)~∞不等价于C_o的通常单位基{e_n}_(n=1)~∞,则存在X上一个等价范数‖.‖,使得关于这个等价范数‖.‖,{x_n}_(n=1)~∞的系数泛函{x_n}_(n=1)~*没有一个Hahn-Banach延拓是一个基序列。文中也提出一个猜测。
In this paper, we give a partial answer for open questions of R. HoIub [1]. We prove that (1)If {x_n}_n~∞=1 is a basic sequence in X for which [x_n]_n~∞=1 is a complemented subspace of X and having coefficient funotionals {x_m~*} in [x_n]~*, then there is an equivalent norm on X for which the sequencce {x_n~*}_n~*=1 has a basic sequence of Hahn Banach extensions in X~*. (2)If {x_n}_n~∞=1 be a bounded basic sequence in a Banach space X for which [x]_N~∞=1 is a eomplemented subspace of X and suppose {x_n}_n~∞=1 is not equivalent to the basis {e_n}_n~∞=1 for c_o, then there is an equivalent norm on X for which no basic sequence of Hahn-banach extensions of the coetticient functionals of {x_n}_n~∞ exists.
出处
《华东师范大学学报（自然科学版）》
CAS
CSCD
1989年第3期29-33,共5页
Journal of East China Normal University(Natural Science)
