Suppose $X$ is a topological vector space and $M$ is a closed linear subspace of $X$. Give $X/M$ the quotient topology induced by the mapping $p:X \to X/M$ defined by $p(x)= x + M$. The show that $X/M$ is a topological vector space, i.e, scalar multiplication and addition are continuous.
For scalar multiplication Assume the field is F. I guess I should look at a digram like this:
$F \times X\xrightarrow {scalar \ \ mult}X$ , then $X \xrightarrow{p} X/M$. then the composition is a continuous map $F \times X\xrightarrow{p \circ .} X/M$. Now I think if the mapping $F \times X \xrightarrow{(id,p)} F \times X/M$ is a quotient map then it induces a continuous map $F \times X/M \to X/M$ that should be the scalar multiplication of $X/M$.
So, the continuity of $+\colon X/M \times X/M \to X/M$ is proved in Rudin's functional book and I give a proof of the continuity of the scalar product using the same idea.
Then to prove that $.:F \times X/M \to X/M$ is contiunous let $w$ be an open neighborhood of $\vec{0}=M$ in $X/M$. $p^{-1}(w)$ is an open neighborhood of $\vec{0}$ in $X$ since $p$ is continuous. Now we use the fact that$.\colon F \times X \to X$ is continuous since $X$ is a TVS. There exists $\delta > 0$ and an open neighborhood $v$ of $\vec{0}$ in $X$ such that $\forall \alpha$ with $\lvert \alpha \rvert <\delta$, $\alpha v \subseteq p^{-1}(w)$. Since $p$ is linear we have $p(\alpha v)= \alpha p(v) \subseteq w$. But we can easily show that $p$ is an open map and thus $p(v)$ is an open neighborhood of $\vec{0}$ in $X/M$. So for every neighborhood $w$ of $\vec{0}$ in $X/M$, $\exists \delta > 0$ and an open neighborhood $p(v)$ of $\vec{0}$ in $X/M$ such that $\forall \alpha$ with $\lvert \alpha \rvert <\delta$, $\alpha p(v) \subseteq w$.