Let $E$ be the normed vector space. Let $X = \cap_i X_i$ be the dense $G_\delta$ subspace. I'm trying to show that in fact each $X_i = E$. If we see that $X_i$ is open, dense and a subspace we are done. Since
From $X_i$ being open it follows that there is a ball $B(y,r) \subset X_i$. Since $X_i$ is dense (follows form X being dense), given any $e \in E$ we have that there is $x\in X_i$ such that $|| e - x|| <r$, and from $X_i$ being a subspace we have that $y - y + x + rB(0,1) \subset X_i$ so $e \in X_i$.
I don't really know if this is a good approach. How would one solve this problem?