Cleanup

src/fa/lc/continuous.tex

@@ -13,11 +13,11 @@
     \end{enumerate}
 \end{proposition}
 \begin{proof}
-    $(1) \Leftrightarrow (2) \Leftrightarrow (3)$: By \ref{definition:continuous-linear}.
+    $(1) \Leftrightarrow (2) \Leftrightarrow (3)$: By \autoref{definition:continuous-linear}.
 
     $(2) \Rightarrow (4)$: $x \mapsto [Tx]_F$ is a continuous seminorm on $E$.
 
-    $(4) \Rightarrow (3)$: Let $U \in \cn_F(0)$ be convex, circled, and radial, then its gauge $[\cdot]_U$ is a continuous seminorm on $F$ by \ref{definition:locally-convex}. Thus there exists a continuous seminorm $[\cdot]_E$ such that $[Tx]_U \le [x]_E$. In which case, $V = \bracs{x \in E| [x]_E < 1} \in \cn_E(0)$ with $T(V) \subset U$. Therefore $T$ is continuous at $0$, and continuous by \ref{definition:continuous-linear}.
+    $(4) \Rightarrow (3)$: Let $U \in \cn_F(0)$ be convex, circled, and radial, then its gauge $[\cdot]_U$ is a continuous seminorm on $F$ by \autoref{definition:locally-convex}. Thus there exists a continuous seminorm $[\cdot]_E$ such that $[Tx]_U \le [x]_E$. In which case, $V = \bracs{x \in E| [x]_E < 1} \in \cn_E(0)$ with $T(V) \subset U$. Therefore $T$ is continuous at $0$, and continuous by \autoref{definition:continuous-linear}.
 \end{proof}
 
 \begin{proposition}
@@ -29,6 +29,7 @@
         \[
         [Tx]_F \le \prod_{j = 1}^n [x_j]_{E_j}
         \]
+
     \end{enumerate}
 \end{proposition}
 \begin{proof}