Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable Banach subspaces $\{X_i\}_{i \in I}$ of $X$ with the property that:

- $D\cap X_i$ is dense in $X_i$,
- $\bigcup_{i \in I} X_i$ is dense in $X$?