I'd just like to know where to begin. The exact thing to prove: Let $X$ and $Y$ be normed spaces and $R:X\to Y$ is an open linear mapping. Show that $R$ is surjective. And to be clear, neither of the assumed normed spaces $X$ and $Y$ are known to be Banach, and they can be infinite-dimensional spaces too.
Hint: If $R$ is open, $R(B_X(0,1))$ contains $B_Y(0,r), r>0$, this implies that $R(B_X(0,n))$ contains $B_Y(0,nr)$. Then use the fact that $\cup_nB_Y(0,nr)=Y$.