Cleanup

src/fa/lc/quotient.tex

@@ -7,6 +7,7 @@
     \[
     \rho_M: E/M \to [0, \infty) \quad x + M \mapsto \inf_{y \in x + M}\rho(y)
     \]
+
     is the \textbf{quotient} of $\rho$ by $M$.
 \end{definition}
 \begin{proof}
@@ -19,6 +20,7 @@
     \[
     \rho_M(x + x') \le \rho(y + y') \le \rho(y) + \rho(y')
     \]
+
     As this holds for all $y \in x + M$ and $y' \in x' + M$, $\rho_M(x + x') \le \rho_M(x) + \rho_M(x')$.
 \end{proof}
 
@@ -36,23 +38,26 @@
         \widetilde E \ar@{->}[r]_{\tilde f} & F
         }
         \]
+
         If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$.
         \item If $\seqi{\rho}$ is a family of seminorms that induces the topology on $E$, then their quotients by $M$ induces the topology on $\td E$.
     \end{enumerate}
     The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$.
 \end{definition}
 \begin{proof}
-    By \ref{definition:tvs-quotient}, (2), (3), (U) holds, and $\td E$ is a TVS over $K$.
+    By \autoref{definition:tvs-quotient}, (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 $E/M$.
 \end{proof}