This question is about complete normed vector space. Let $X$ be such space, $\{v_{n}\}_{n\in\mathbb{N}}$ be a linear independent set. Let $x\in X$ is in the form of $\sum_{n=1}^{\infty}\sum_{k=1}^{n}a_{nk}v_{k}$, $a_{nk}\in\mathbb{C}$. Is the coefficient of $v_{n}$ well defined? i.e. We can rearrange $a_{nk}$ so that $\sum_{k=1}^{\infty}a_{kn}$ converge?
Clearly if the series is absolute converge, it doesn't matter, how about conditional convergent?