Is a quotient of topological vector space a topological vector space

Question

I should prove or disprove that quotient of topological vector space is a topological vector space.
I think that it is. May I reduce my prove to proof of following statement?
$X$ - topological vector space, $M \subset X$ - linear subspace of $X$. Let $A$ - vector addition in $X/M$, W - neighbourhood of origin $o$ in $X/M$. Then i should prove that $A^{-1}(W)$ is a neighbourhood of $(o, o)$ in $X/M \times X/M$.
But I have no idea how can I prove this fact.

Answer 1

A neighborhood of $0$ in $X/M$ is any set $W$ containing $0+M$ such that $\pi^{-1}(W)$ is a neighborhood of $0$ in $X$, where $\pi\colon X\to X/M$ is the canonical projection.

Let $W$ be a neighborhood of $0$ in $X/M$. Then there exists a neighborhood $V$ of $0$ in $X$ such that $V+V\subseteq \pi^{-1}(W)$.

This implies that $\pi(V)+\pi(V)\subseteq W$. Since $V\subseteq\pi^{-1}(\pi(V))$, we have that $\pi(V)$ is a neighborhood of $0$ in $X/M$. This proves continuity of the addition in $X/M$.