Let $X$ be a separable locally convex space and $D\subseteq X$ be dense. Then, is the set of all finite-linear combinations of $D$ equal to $X$?
That is, when is $\operatorname{span}(D) = cl(D)$?
2026-03-25 12:44:51.1774442691
Linear Combinations of a Dense Set
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It is very easy to find dense linear subspaces $M$ of a normed linear space which are proper subsets of $X$. Example, $\ell^{1}$ is a proper linear subspace of $\ell^{2}$ and it is dense. So the answer is NO.