If $(M_{\lambda})$ be a chain of closed affine subspaces of $X$ $\Longrightarrow$ $\displaystyle \bigcap_{\lambda\in L} M_{\lambda}\neq \emptyset \;$?

Question

Let $X$ be a Banach space

Let $(M_{\lambda})_{\lambda \in L}$ be a chain of closed affine subspaces of $X$

We can say that $$\displaystyle \bigcap_{\lambda\in L} M_{\lambda}\neq \emptyset \;\;?$$

affine subspace: translate of a linear subspace

chain: $\lambda_1 \neq \lambda_2$ , $M_{\lambda_1}\subset M_{\lambda_2}$ or $M_{\lambda_2}\subset M_{\lambda_1}$

Any hints would be appreciated.

Answer 1

Hint: consider $X=\ell_1$, $L=\mathbb N$, and $$M_n=\left\{(x_k)\in \ell_1: x_1=\dots=x_n=0, \ \sum_{k\in\mathbb N} x_k=1\right\},\quad n\in\mathbb N$$