TITLE: Subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?
QUESTION [7 upvotes]: Are there any examples of subspaces  of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?

REPLY [3 votes]: As an observation, every subspace generated by countably many linearly independent vectors (that is, a subspace of dimension $\aleph_0$) inside a Banach space is not closed, since a Banach space cannot have (algebraic) dimension $\aleph_0$. ( if it were closed, if would be itself a Banach space).