Let $X$ be Banach, and $Y \subset X$ a (strict) linear subspace with the property that for any $f,g \in X^*$, if $f \neq g$ then $f|_Y \neq g|_Y$. What can we say about the density of $Y$ in $X$?
Any help is appreciated - I saw this question online and it isn't for homework. I'm trying to use Hanh-Banach to no avail.
If $Y$ was not dense in $X$, then, by Hahn-Banach, for any $x\notin\overline Y$ we can always find some $f\in X^*$ such that $f|\overline Y = 0$ and $f(x) = 1$. So, obviously, $f\neq g:=0$. The assumption now says that also $f|Y\neq g|Y$, which is obviously false. Hence, $Y$ must be dense in $X$.