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

src/fa/lc/bornologic.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item For any $U \subset E$ convex and balanced, if $U$ absorbs every bounded set of $E$, then $U \in \cn_E(0)$.
         \item For any seminorm $\rho: E \to [0, \infty)$ that is bounded on all bounded sets of $E$, $\rho$ is continuous. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     If the above holds, then $E$ is a \textbf{bornological space}.
 \end{definition}

src/fa/lc/convex.tex

@@ -138,7 +138,8 @@
         \item For each $i \in I$, $d_i: E \times E \to [0, \infty)$ defined by $(x, y) \mapsto [x - y]_i$ is a pseudo-metric.
         \item The topology induced by $\seqi{d}$ makes $E$ a topological vector space.
         \item For each $i \in I$, $[\cdot]_i: E \to [0, \infty)$ is continuous.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The topology induced by $\seqi{d}$ is the \textbf{vector space topology induced by} $\seqi{[\cdot]}$. In addition,
     \begin{enumerate}
         \item[(U)] For any family $\bracsn{[\cdot]_j}_{j \in J}$ of continuous seminorms on $E$, the vector space topology induced by $\bracsn{[\cdot]_j}_{j \in J}$ is contained in the vector space topology induced by $\seqi{[\cdot]}$.
@@ -159,7 +160,8 @@
         \item If $A$ is convex, then for any $x, y \in E$, $[x + y]_A \le [x]_A + [y]_A$.
         \item If $A$ is circled, then for any $x \in E$ and $\lambda \in K$, $[\lambda x]_A = \abs{\lambda}[x]_A$.
         \item If $A$ is circled, then $\bracs{\rho < 1} \subseteq A \subseteq \bracs{\rho \le 1} \subseteq \ol A$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular,
     \begin{enumerate}[start=4]
         \item If $A$ is convex, then $[\cdot]_A$ is a sublinear functional.
@@ -189,7 +191,8 @@
         \item There exists a fundamental system of neighborhoods at $0$ consisting of convex sets.
         \item There exists a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets.
         \item There exists a family of seminorms $\seqi{[\cdot]}$ that induces the topology on $E$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $E$ is a \textbf{locally convex} space.
 \end{definition}
 \begin{proof}

src/fa/lc/hahn-banach.tex

@@ -134,7 +134,8 @@
         \begin{enumerate}
             \item $|\phi| \le \rho$.
             \item $\dpb{x, \phi}{E} = \inf_{y \in M}\rho(x + y)$.
-        \end{enumerate}
+        \end\{enumerate\}
+
         \item For any $x \in E$ and continuous seminorm $\rho: E \to [0, \infty)$, there exists $\phi \in E^*$ with $|\phi| \le \rho$ and $\dpb{x, \phi}{E} = \rho(x)$.
         \item If $E$ is separated, then for any $x, y \in E$, there exists $\phi \in E^*$ with $\dpb{x, \phi}{E} \ne \dpb{y, \phi}{E}$.
     \end{enumerate}

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 $\lambda (U \cap E_n) \supset B$.

src/fa/lc/quotient.tex

@@ -41,7 +41,8 @@
 
         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}
+    \end\{enumerate\}
+
     The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$.
 \end{definition}
 \begin{proof}

src/fa/lc/spaces-of-linear.tex

@@ -6,7 +6,8 @@
     Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, $E$ be a locally convex space over $K$, and $\cf \subset E^T$ be a subspace such that
     \begin{enumerate}
         \item[(B)] For each $f \in \cf$ and $S \in \sigma$, $f(S) \subset E$ is bounded. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let
     \[

src/fa/lc/tensor.tex

@@ -23,7 +23,8 @@
         \]
 
         is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the \textbf{canonical embedding}.
 
@@ -72,7 +73,8 @@
     and the seminorm $\rho = p \otimes q$ is the \textbf{cross seminorm} of $p$ and $q$. Moreover, 
     \begin{enumerate}
         \item[(5)] If the seminorms $\seqi{p}$ define the topology on $E$, and the seminorms $\seqj{q}$ define the topology on $F$, then the seminorms $\bracsn{p_i \otimes q_j| (i, j) \in I \times J}$ define the topology on $E \otimes_\pi F$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{definition}
 \begin{proof}[Proof {{\cite[III.6.3]{SchaeferWolff}}}. ]
@@ -140,7 +142,8 @@
         \item $\sum_{n \in \natp}|\lambda_n| < \infty$. 
         \item $\limv{n}x_n = 0$ and $\limv{n}y_n = 0$.
         \item $z = \sum_{n = 1}^\infty \lambda_n x_n \otimes y_n$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{theorem}
 \begin{proof}
@@ -162,7 +165,8 @@
         \item $v_N = \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k} \otimes y_{N, k}$.
         \item For each $1 \le k \le n_N$, $p_N(x_{N, k}), q_N(x_{N, k}) \le 1/M$.
         \item $\sum_{k = 1}^{n_N}|\lambda_k| \le 2^{-N+2}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     From here, let $\seqf{(x_j, y_j)} \subset X \times Y$ such that $u_1 = \sum_{j = 1}^n x_j \otimes y_j$, then
     \[