# Intersection of spaces with Schauder basis

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:

$$\bigcap_{m=1}^\infty \left( V_m + U_m \right) = \{0\}$$

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 $$1\le p there exists $$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.$$
• (Here $\mathbb{N}=\{1,2,3,\dots\}$) – Pietro Majer Apr 17 '19 at 22:11