Breaking up dense subset in non-separable space

Question

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 $X$?

Answer 1

For every countable subset $M\subseteq D$ set $X_M=\overline{\text{span}(M)}$ (with the norm of $X$). These spaces seem to satisfy your requirements. Am I missing something?