数学物理学报(英文版) ›› 2003, Vol. 23 ›› Issue (3): 351-.
杨长森
YANG Chang-Sen
摘要:
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.
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