Some Equivalent Characterizations And Related Properties Of Unconditional Convergence Of Infinite Series In Banach Space

Infinite series has been playing an irreplaceable role in the development of mathematics.The theory of infinite series in Banach space is the generalization of number series,and the unconditional convergence is an important kind of convergence properties of infinite series in Banach space.Starting from the unconditional Cauchy property of series,this paper studied and illustrated in details about the relationships among the unconditional convergence,the subsequence convergence,the bounded multiplier convergence,the rearrangement convergence and the Symbolic Convergence of the intermediate number in the normed space under norm topology.We also pointed out the equivalence of the above convergence in the Banach space,discussed the related properties of the unconditional convergence series,and briefly explained the relationship between the absolute convergence and the unconditional convergence.Two of them discuss other equivalent characterizations and related properties of unconditional Cauchy series.The unconditional convergence of intermediate number is closely related to the absolute convergence of intermediate number in Banach space,in which the absolute convergence implies the convergence.Similarly,it can be shown that absolute convergence also implies unconditional convergence.We have known that the absolute convergence of series with number terms and rearrangement convergence of series with number terms are equivalent.which could be extended to the normed space of any finite dimension.However,it is not necessarily in infinite dimensional space.As for the unconditional and absolute convergence of intermediate numbers in infinite dimensional space,there are series with unconditional but not absolute convergence in every infinite dimensional Banach space.In addition,this paper studied the unconditional convergence of infinite series in Banach space under weak topology,and completely supplemented,the five weak convergence relations in Banach space,including the definition of the above five kinds of convergence in Banach space under weak topology and the complete proof of their relationships.Different from norm topology,the convergence of the subsequence of the intermediate number is strictly stronger than unconditional convergence in Banach space under weak topology.