Let $(X,\|\cdot\|_1)$ be a separable Banach space and $(X',\|\cdot\|_2)$ be a dense subspace such that $\|x\|_1 \le C \|x\|_2$ holds for a constant $C>0$ and every $x \in X'.$
I know that the two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ are not necessarily equivalent. However, I was wondering if there exists a set $X'' \subseteq X'$ (see below) such that $\|x\|_1 \ge C' \|x\|_2$ goes for some constant $0<C'<C$ and every $x \in X''.$
What I would like to show is that within every open ball (with respect to $\|\cdot\|_1$) around some point in $X$ there lies some $x \in X'$ such that $C' \|x\|_2 \le\|x\|_1 \le C \|x\|_2.$
Are there any easy counterexamples to this that I am not aware of?