Definition 8.6.1 (Quotient TVS). Let $E$ be a TVS over $K \in \RC$, and $M \subset E$ be a vector subspace, then there exists $(\td E, \pi)$ such that:
$\td E$ is a TVS over $K$.
$\pi \in L(E; \td E)$.
$\ker \pi \supset M$.
For any topological space $F$ and $f \in C(E;F)$ such that $f(x) = f(y)$ whenever $x - y \in M$, there exists a unique $\td f \in C(\td E; F)$ such that the following diagram commutes
\[\xymatrix{ E \ar@{->}[rd]^{f} \ar@{->}[d]_{\pi} & \\ \widetilde E \ar@{->}[r]_{\tilde f} & F }\]If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$.
The space $\td E = E/M$ is the quotient of $E$ by $M$.
Proof. Let $\td E = E/M$ be the algebraic quotient of $E$ by $M$, and equip it with the quotient topology by $\pi$.
(1): By Definition 4.11.3, for each $\pi(U) \subset E/M$, $\pi(U)$ is open if and only if $U$ is open. Since the topology on $E$ is translation-invariant, so is the quotient topology on $E/M$. Let
Since the circled and radial neighbourhoods of $0$ forms a fundamental system of neighbourhoods at $0$ by Proposition 8.1.11, $\fB$ is a fundamental system of neighbourhoods at $0$ for the quotient topology on $E/M$. In addition,
Let $U \in \cn(0)$ be circled and radial. For any $\lambda \in K$ with $\abs{\lambda}\le 1$, $\lambda \pi(U) = \pi(\lambda U) \subset \pi(U)$, so $\pi(U)$ is also circled. For any $x + M \in E/M$, there exists $\lambda \in K$ such that $x \in \lambda U$. In which case, $x \in \lambda U + M = \pi(U)$, so $\pi(U)$ is also radial.
For any $U \in \cn(0)$ circled and radial, by Proposition 8.1.11, there exists $W \in \cn(0)$ such that $W + W \subset U$. In which case, $\pi(W) + \pi(W) \subset \pi(U)$.
By Proposition 8.1.12, there exists a unique translation-invariant topology on $E/M$ such that $\fB$ is a fundamental system of neighbourhoods at $0$, which must be the quotient topology on $E/M$. In which case, the quotient topology is a vector space topology by (3) of Proposition 8.1.12.
(2), (3), (U): By Definition 4.11.3.$\square$