Banach空间中级数弱无条件收敛性的若干等价刻画

Equivalent Characterizations of Weak Unconditional Convergence of Infinite Series in Banach Spaces

阐述了范数拓扑下赋范空间中无穷级数的无条件收敛性、子列收敛性、有界乘子收敛性、重排收敛性和符号收敛性及对应的Cauchy性质的定义及其之间的关系,回顾了级数绝对收敛性与无条件收敛性的关系,阐述了上述5种收敛性在弱拓扑下的Banach空间中的定义,给出了其相互关系的完整证明,比较了与范数拓扑下的异同。

In the case of Banach spaces under norm topology,definitions of unconditional convergence,subseries convergence,bounded multiplier convergence,reordered convergence and sign convergence and the corresponding Cauchy properties of infinite series were stated.The relationship among the convergences above and the relationship between unconditional convergence and absolute convergence were revisited.The counterparts of the five convergences in Banach spaces under weak topology were provided,as well as a complete proof of their relationship.Similarities of and differences between norm and weak topology were compared.