Use the Hahn-Banach theorem to show that if every bounded linear map $T:X_{1} \rightarrow X_{2}$ is surjective, then $\dim X_{2}=1$. I found the following question in my textbook:
Let $X_{1}$ and $X_{2}$ normed spaces (both nontrivial). Use the Hahn-Banach theorem to show that if every bounded linear map $T:X_{1} \rightarrow X_{2}$ is surjective, where $T \neq 0$, then $\dim X_{2}=1$.

At the time I had few ideas, but i can't solve it. If someone help me, I'll be grateful.

Hahn-Banach implies that for every $z\in X_1$ there exists a continuous linear functional $\varphi$ from $X_1$ to the scalars, having norm 1 and mapping $z$ to its own norm.
Pick any nonzero $z\in X_1$ and nonzero $y\in X_2$. Then $x\mapsto \varphi(x)y$ is a continuous linear mapping that only hits the multiples of $y.$ It cannot be surjective unless $X_2$ is generated by $y.$
