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:


$$
\bigcap\_{m=1}^\infty
\left(
V\_m + U\_m
\right)
=
\{0\}
$$



Intersection of spaces with Schauder basis
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 $U\_m+V\_m\supset \operatorname{span}(v\_1,\dots,v\_{m-1})+V\_m=\ell\_2.$


