级数的绝对收敛性问题

The Problem on the Absolute Convergence of Series

阐述了赋范线性空间中无穷级数的收敛、绝对收敛、无条件收敛等概念之间的关系,并例证说明级数的收敛与绝对收敛、绝对收敛与无条件收敛之间不等价,但确实存在着无穷维的Fréchet空间中级数的无条件收敛与绝对收敛等价。

This article mainly introduces the relationship between convergent series,absolutely convergent series and unconditionally convergent series in normed linear spaces.And through some counterexamples we draw the important conclusions that convergent series is unequal to absolute convergence and also the unconditional convergence is unequal to absolute convergence.This article also proves that absolute convergence is equivalent to unconditional convergence in infinite dimensional Fréchet space.