Is $u=\sum_{n=1}^{\infty}\frac{1}{2^{n}}e_n$ an element of X, where $\{e_{n}\}_{n=1}^{\infty}$ is a maximal set of linearly independent vectors in X and X is a Banach space?
In other words, are the partial sums $\sum_{n=k}^{m}\frac{1}{2^{n}}e_n$ a Cauchy sequence? If yes, then by completeness $u\in X$.
Thanks
This need not be true.
Consider the following vectors in $l_\infty$. $e_n=(2^n,0,...0,1,0,...)$. These vectors are certainly in $l_\infty$ and are certainly linearly independent. What happens when you try to add them up (even after normalization)?