Intersection of spaces with Schauder basis

Question

Let $\{v_n\}_{n \in \mathbb{N}}$ be a basic sequence in $\ell^2$ over $\mathbb{C}$

Let $V_m =\overline{\operatorname{span}} \{v_n\}_{n \geq m} $

Let $\{u_n\}_{n \in \mathbb{N}}$ be a basic sequence in $\ell^2$ over $\mathbb{C}$

Let $U_m =\overline{\operatorname{span}} \{u_n\}_{n \geq m} $

Let $W_m = V_m + U_m$

I would like to know if it is true that

$$ \bigcap_{m=1}^\infty W_m = \{0\} $$

Thanks.

Answer 1

Let $(e_n)$ be the standard unit vectors in $\ell_2$.

Choose $(\alpha_n)\in\ell_2$ with $\alpha_n\ne0 $ for each $n$ to have $\ell_2$ norm so small that the sequence $$e_1,\ \alpha_1 e_1+e_2,\ \alpha_2 e_1+e_3,\ \ldots$$ is basic (you can do this using perturbation results for basic sequences). Take this to be $(v_n)$ and take $(u_n)$ to be $(e_n)$ . Then $e_1\in W_n$ for all $n$.