摘要
设正数序列w=(wn)满足1=w1≥w2≥…≥wn≥wn+1…,limn→∞wn=0,∑∞n=1wn=∞.对任何Banach空间序列{Xn},定义Banach空间值Lorentz序列空间X为X=d1(w,{Xn})={(xn):xn∈Xn,(xn)=supπ∑∞n=1wnxπ(n)<+∞}其中π取遍所有正整数集的置换.证明弱序列完备和遗传地含有l1这两个性质可以从{Xn}遗传到X上.但是X是强弱紧生成空间的充要条件是每个Xn是强弱紧生成空间,并且除有限个之外的所有Xn都是自反空间.也给出一个弱序列完备并遗传地含有l1但不是强弱紧生成的可分Banach空间,从而否定地回答了文献[1]中的一个公开问题.最后给出具基Banach空间是强弱紧生成空间的一些等价条件.
Let w=(wn) be a sequence of positive numbers with 1=w1≥w2≥…≥wn≥wn+1…,limn→∞wn=0,∑∞n=1wn=∞.For any sequence {Xn} of Banach spaces,define the Banach space_valued Lorentz sequence space X byX=d1(w,{Xn})={(xn):xn∈Xn,(xn)=supπ∑∞n=1wnxπ(n)<+∞}where π ranges over all permutations of the positive integers.We prove that weak sequential completeness as well as containing l1 hereditarily can be passed from {Xn} to X respectively.However,X is strongly weakly compactly generated(SWCG) if and only if all Xn are SWCG and all but finitely many of the Xn's are reflexive.We also give an example of a separable and weakly sequentially complete Banach space containing l1 hereditarily which is not SWCG,answering an open question in [1] negatively.Finally,some equivalent conditions are given for Banach spaces with symmetric or subsymmetric basis to be SWCG.
出处
《鞍山科技大学学报》
2003年第4期246-253,共8页
Journal of Anshan University of Science and Technology
关键词
Banach空间值
Lorentz序列空间
弱序列完备
强弱紧生成空间
自反空间
Banach space-valued Lorentz sequence space
weakly sequentially complete
containing l^1 hereditarily
strongly weakly compactly generated space
reflexive space
