TITLE: Intersection of spaces with Schauder basis
QUESTION [5 upvotes]: Let $\{v_n\}_{n \in \mathbb{N}}$ be a Schauder basis of $V$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $V_m = \overline{\operatorname{span}} \{v_n\}_{n \geq m}$
Let $\{u_n\}_{n \in \mathbb{N}}$ be a Schauder basis of $U$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $U_m = \overline{\operatorname{span}} \{u_n\}_{n \geq m}$
Under the hypothesis that $\forall m \in \mathbb{N}: V_m + U_m$ is closed, is it true that:
$$
\bigcap_{m=1}^\infty
\left(
V_m + U_m
\right)
=
\{0\}
$$

REPLY [6 votes]: In general it is not true: $V_m$ and $U_m$ could even be transverse for all $m$, giving
$$
\bigcap_{m=1}^\infty
\left(
V_m + U_m
\right)
=
\ell_2.
$$
Let $\kappa:\mathbb{N}\to\mathbb{N}$ be a map such that every $p\in\mathbb{N}$ has fiber $\kappa^{-1}(p)$ of infinite cardinality. 
Let $\{v_n\}_{n\ge1}$ be the orthonormal basis of $\ell_2$ and let $K:\ell_2\to\ell_2$ be the bounded linear operator defined by  $Kv_n= 2^{-n}v_{\kappa(n)}$ for all $n\ge1$. Then
$$\|K\|\le\|K\|_{HS}=\Big(\sum_{n\ge1}\|Kv_n\|^2\Big)^{1/2}=\Big(\sum_{n\ge1}4^{-n}\Big)^{1/2}=3^{-1/2}<1$$
Therefore $I+K$ is invertible and $u_n:=(I+K)v_n$ for $n\ge1$ defines a Schauder basis.
However, by the hypothesis on $\kappa$, for any $1\le p<m$  there exists $n\ge m$ such that $\kappa(n)=p$, so $u_n=v_n+2^{-n}v_p\in U_m$, and $v_p=2^nu_n-2^nv_n\in U_m+V_m$, hence $U_m+V_m\supset \operatorname{span}(v_1,\dots,v_{m-1})+V_m=\ell_2.$