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 $X$?

1 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?

  • 2
    $\begingroup$ No i think I was missing something... Thank you. $\endgroup$
    – ABIM
    Commented Jun 10, 2020 at 8:43

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