Polished A-A and added new lines for broken enumerates.

src/fa/lc/inductive.tex

@@ -21,7 +21,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     The topology $\topo$ is the \textbf{inductive locally convex topology} on $E$ induced by $\seqi{T}$.
 \end{definition}
 \begin{proof}
@@ -64,7 +65,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The space $E = \bigoplus_{i \in I}E_i$ is the \textbf{locally convex direct sum} of $\seqi{E}$. 
 \end{definition}
@@ -110,7 +112,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the \textbf{inductive limit} of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$.
 \end{definition}
 \begin{proof}
@@ -174,7 +177,8 @@
         \begin{enumerate}
             \item[(a)] $B$ is bounded.
             \item[(b)] There exists $n \in \natp$ such that $B \subset E_n$ is bounded.
-        \end{enumerate}
+        \end\{enumerate\}
+
         \item If $E_n$ is complete for each $n \in \natp$, then $E$ is also complete.
     \end{enumerate}
 \end{proposition}
@@ -190,7 +194,8 @@
         \item For each $k \in \natp$, $U_k \in \cn_{E_{n_k}}(0)$.
         \item For each $k \in \natp$, $U_k = U_{k+1} \cap E_{n_k}$.
         \item For each $k \in \natp$, $n^{-1}x_k \not\in U_k$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $V = \bigcup_{k \in \natp}U_k \in \cn_E(0)$ with $V \cap E_{n_k} = U_k$ for all $k \in \natp$. For any $n \in \natp$, $x_k \not\in nU_k = nV \cap E_{n_k}$. Therefore $B$ is not bounded.
 
     (3), $(b) \Rightarrow (a)$: Let $U \in \cn_E(0)$, then $U \cap E_n \in \cn_{E_n}(0)$, so there exists $\lambda \in K$ with $\lambda (U \cap E_n) \supset B$.