A question on linearly independent vectors in a Banach space Given a list of linearly independent vectors $\{x_1,...,x_n\}$ in a Banach space. If for each $1 \leq i\leq n$, there is a sequence of vectors $\{y_m^{(i)}\}_{m=1}^{\infty}$ converges to $x_i$. Then when $m$ is big enough, whether $\{y_m^{(i)}:1 \leq i\leq n\}$ is also linearly independent?

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Yes. Suppose for the sake of convenience that the $y$'s are dependent for every $m$. Choose $c_m^i$ with $$\sum_{i=1}^nc_m^iy_m^i=0,$$and, say, $$\sum_{i=1}^n|c_m^i|=1.$$Now the vectors $c_m\in\Bbb R^n$ have a non-zero accumulation point...
This is not true, even in the simplest case of $\mathbb{R}^2$.  Take $x_1, x_2$ as the standard basis, $y^{(1)}_m=(1,1/m)$, $y^{(2)}_m=(1/m,1)$.