关键词: Basic sequence, Multiple series

Abstract:

Let $X$ be a Banach space and $\{e_j\}^{\oo}_{j=1}$ be a sequence
 in $X$. The author shows
 that $\{e_j\}^{\oo}_{j=1}$ is a basic sequence if and only if
$\sum^{\oo}_{n=1}r_n \a_{nj}$  converges for every  $j\>1$ and
$\sum^{\oo}_{n=1}r_n\sum^{\oo}_{j=1}\a_{nj}e_j= \sum^{\oo}_{j=1}(\sum^{\oo}_{n=1}r_n\a_{nj})e_j$
holds for every choice of scalar variables $\{\a_{nj}\}$
such that $\sum^{\oo}_{j=1}\a_{nj}e_j$ converges for each
 $n\>1$
 and any choice of scalar variables $\{r_n\}$ such that $\sum^{\oo}_{n=1} \sum^{\oo}_{j=1}r_n\a_{nj}e_j$ converges.
 Moreover,  some applications about the result are given.

Key words: Basic sequence, Multiple series