Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-empty open subset of $X$. But how to show this?



Dense linear span implies closed convex hull has non-empty interior
The answer is no. E.g., let $X=\ell^2$ and 
$$Y=\{x\in\ell^2\colon\sum\_n nx\_n^2\le1\}.$$
Then $Y$ is a closed convex set, spanning $X$, but the interior of $Y$ is empty. 




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**Details:** $Y$ is convex because the function $\mathbb R\ni u\mapsto u^2$ is convex. To show that $Y$ is closed one may use the Fatou lemma. 


If the interior of $Y$ were not empty, then, by the symmetry of $Y$ (that is, by the property $-Y=Y$), $0$ would be in the interior of $Y$. So, for some real $h>0$ and all natural $n$, we would have $he\_n\in Y$, where $(e\_1,e\_2,\dots)$ is the standard basis of $\ell^2$. But $he\_n\notin Y$ for $n>1/h^2$, a contradiction. 


