Lets $X$ be a non-normable topological vector space and let $Y\subset X$ be a proper subspace. Clearly if $Y$ is dense in $X$ then $Y$ must be non-normable too. Can we have that conclusion with weaker hypothesis? I mean, the non-normability of the ambient space implies the non-normability of a subspace for some weaker hypothesis that density (about the subspace)?
Thaks in advance