The Problem of Convergence of Infinite Series in Banach Space

In normed space with limited dimension and infinite dimensional Freche space,unconditional convergence of series is equivalent with absolute convergence.This paper discusses the relations of several concepts such as convergence,absolute convergence,conditional convergence,unconditional convergence and weak unconditional convergence of series in normed spaces with infinite dimension.And using some counterexamples we explained the important conclusions that weakly unconditional convergent series is unnecessary convergent and unconditional convergent series is unnecessary absolute convergent.