Definition 9.3.2 (Quotient Locally Convex Space). Let $E$ be a locally convex space over $K \in \RC$ and $M \subset E$ be a vector subspace, then there exists $(\td E, \pi)$ such that:
$\td E$ is a locally convex space 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 $\td f \in L(E; F)$.
If $\seqi{\rho}$ is a family of seminorms that induces the topology on $E$, then their quotients by $M$ induces the topology on $\td E$.
The space $\td E = E/M$ is the quotient of $E$ by $M$.
Proof. By Definition 8.6.1, (2), (3), (U) holds, and $\td E$ is a TVS over $K$.
(1): Let $U \subset E$ be convex, then for any $x + M, y + M \in \pi(U)$ and $t \in [0, 1]$,
so $\pi(U)$ is convex. Let $\fB = \bracs{U|U \in \cn_E(0) \text{ convex}}$, then $\bracs{\pi(U)|U \in \fB}$ is a fundamental system of neighbourhoods for the quotient topology on $E/M$. Therefore $E/M$ is locally convex.
(5): By (U), each quotient seminorm is continuous on $\td E$, so the quotient topology contains the topology induced by the quotient seminorms. On the other hand, let $\pi(U) \in \cn_{\td E}(0)$, then there exists $J \subset I$ finite and $r > 0$ such that
For each $j \in J$, let $\eta_{j}$ be the quotient of $\rho_{j}$ by $M$. Let $x + M \in E/M$ with $\eta_{j}(x) < r$ for all $j \in J$. For each $j \in J$, there exists $y_{j} \in x + M$ such that $\rho_{j}(y_{j}) < r$, so $y_{j} + M \in \pi(U)$. Therefore $x \in \pi(U)$ as well, and the quotient norms induce the quotient topology on $E/M$.$\square$