9.3 Quotient Spaces

Definition 9.3.1 (Quotient Seminorm). Let $E$ be a vector space over $K \in \RC$, $\rho: E \to [0, \infty)$ be a seminorm, and $M \subset E$ be a vector subspace, then

\[\rho_{M}: E/M \to [0, \infty) \quad x + M \mapsto \inf_{y \in x + M}\rho(y)\]

is the quotient of $\rho$ by $M$.

Proof. Let $\lambda \in K$ with $\lambda \ne 0$, then $x \mapsto \lambda x$ is a bijection, and

\begin{align*}\abs{\lambda}\rho_{M}(x)&= \abs{\lambda}\inf\bracs{\rho(y)|y \in x + M}= \inf\bracs{\abs{\lambda}\rho(y)|y \in x + M}\\&= \inf\bracs{\rho(\lambda y)|y \in x + M}=\inf\bracs{\rho(y)|y \in \lambda(x + M)}= \rho_{M}(\lambda(x + M))\end{align*}

For any $x, x' \in E$, $y \in x + M$ and $y' \in x' + M$, $y + y' \in x + x' + M$, so

\[\rho_{M}(x + x') \le \rho(y + y') \le \rho(y) + \rho(y')\]

As this holds for all $y \in x + M$ and $y' \in x' + M$, $\rho_{M}(x + x') \le \rho_{M}(x) + \rho_{M}(x')$.$\square$

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 $t \in [0, 1]$,

\[(tx + M) + ((1 - t)y + M) = (tx + (1-t)y) + M \in U + M = \pi(U)\]

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

\[\bigcap_{j \in J}B_{j}(0, r) \subset U\]

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 $E/M$.$\square$

Direct References

Definition 8.6.1: Quotient TVS

Jerry's Digital Garden

9.3 Quotient Spaces

Direct References

Jerry's Digital Garden

Direct References