This is just to make Nate Eldridge's answer selfcontained.
For any normed vector space $V$ and any $r > 0$, write $V_r$ for the open ball of radius $r$ and center $0$ in $V$.
Let $X$ be a normed vector space, $Y$ a closed space, $Z$ the quotient, $\pi$ the canonical projection, $\tau$ the quotient topology, $\nu$ the topology on $Z$ induced by the quotient norm $$|\pi(x)|:=\inf_{y\in Y}|x+y|.$$
(It's easy to see that this is a norm.)
We claim $\tau=\nu$.
Both topologies are translation invariant.
The set $\{\pi(X_r)\ |\ r > 0\}$ is a basis for the $\tau$-neighborhoods of $0$ in $Z$.
The set $\{(Z_r)\ |\ r > 0\}$ is a basis for the $\nu$-neighborhoods of $0$ in $Z$.
As $\pi(X_r)=Z_r$, we're done.
