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

src/fa/duality/polar.tex

@@ -27,7 +27,8 @@
     \begin{enumerate}
         \item $\emptyset^\circ = \emptyset^\square = F$ and $F^\circ = F^\square = \bracs{0}$. 
         \item For any $A, B \subset E$ and $\lambda \ne 0$, if $\lambda A \subset B$, then $B^\circ \subset \lambda^{-1}A^\circ$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 

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
     \[

src/fa/lp/definition.tex

@@ -107,7 +107,8 @@
     \begin{enumerate}
         \item[(a)] $f_n \to f$ strongly pointwise.
         \item[(b)] There exists $g \in L^p(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then $f_n \to f$ in $L^p(X; E)$.
 \end{proposition}

src/fa/norm/absolute.tex

@@ -12,7 +12,8 @@
     \begin{enumerate}
         \item $E$ is a Banach space.
         \item For any absolutely convergent series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$, there exists $x \in E$ such that $x = \sum_{n = 1}^\infty x_n$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{lemma}
 \begin{proof}
@@ -44,7 +45,8 @@
         \item If $\sum_{n \in P}x_n = \infty$ but $\sum_{n \in N}x_n > -\infty$, then $\sum_{n = 1}^\infty x_n$ converges to $\infty$ unconditionally.
         \item If $\sum_{n \in N}x_n = -\infty$ but $\sum_{n \in P}x_n < \infty$, then $\sum_{n = 1}^\infty x_n$ converges to $-\infty$ unconditionally. 
         \item If $\sum_{n \in \natp}|x_n| < \infty$, then $\sum_{n = 1}^\infty x_n$ converges unconditionally.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In other words, a series in $\real$ converges unconditionally if and only if its positive parts or its negative parts are finite. 
 \end{theorem}
 \begin{proof}

src/fa/norm/multilinear.tex

@@ -8,7 +8,8 @@
         \item For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$.
         \item For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$.
         \item $E$ is a Banach space.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $T \in L^2(E, F; G)$.
 \end{proposition}
 \begin{proof}

src/fa/norm/normed.tex

@@ -38,12 +38,14 @@
     \begin{enumerate}
         \item[(a)] $\norm{x}_E \le C\norm{y}_F$.
         \item[(b)] $\norm{y - Tx}_F \le \gamma \norm{y}_F$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that:
     \begin{enumerate}
         \item $\sum_{n \in \natp}\norm{x_n}_E \le C\norm{y}_F/(1 - \gamma)$.
         \item $\sum_{n = 1}^\infty Tx_n = y$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular, if $E$ is a Banach space, then for every $y \in F$, there exists $x \in E$ such that $\norm{x}_E \le C\norm{y}_F/(1 - \gamma)$ and $Tx = y$.
 \end{theorem}
 \begin{proof}
@@ -73,7 +75,8 @@
     \begin{enumerate}
         \item For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$.
         \item $E$ is a Banach space.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)} < \infty$.
 \end{theorem}
 \begin{proof}

src/fa/norm/separable.tex

@@ -29,7 +29,8 @@
         \item $\bracs{B(x, r)|x \in E, r > 0}$.
         \item $\bracsn{\ol{B(x, r)}|x \in E, r > 0}$. 
         \item Open sets in $E$ with respect to the weak topology. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}

src/fa/order/lattice.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item[(LO1)] For any $x, y, z \in E$ with $x \le y$, $x + z \le y + z$.
         \item[(LO2)] For any $x, y \in E$ and $\lambda > 0$, $x \le y$ implies that $\lambda x \le \lambda y$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{definition}
 
@@ -17,7 +18,8 @@
     \begin{enumerate}
         \item $\sup(A + B) = \sup(A) + \sup(B)$.
         \item $\sup(A) = -\inf (-A)$
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}
@@ -103,12 +105,14 @@
     \begin{enumerate}
         \item[(3)] $|\lambda x| = |\lambda| \cdot |x|$
         \item[(4)] $|x + y| \le |x| + |y|$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     Finally, for any $x, y \in E$ with $x, y \ge 0$, 
     \begin{enumerate}
         \item[(5)] $[0, x] + [0, y] = [0, x + y]$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}
@@ -175,7 +179,8 @@
     \begin{enumerate}
         \item For any $x \in C$ and $\lambda \in \real$ with $\lambda \ge 0$, $\phi(\lambda x) = \lambda \phi(x)$.
         \item For any $x, y \in C$, $\phi(x + y) = \phi(x) + \phi(y)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then the mapping
     \[
@@ -218,7 +223,8 @@
         \[
         |\phi|(x) = \sup\bracs{\phi(y)|y \in E, |y| \le x}
         \]
-    \end{enumerate}
+    \end\{enumerate\}
+
     
 \end{proposition}
 \begin{proof}

src/fa/rs/bv.tex

@@ -67,7 +67,8 @@
         \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$.
         \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}: E^{[a, b]} \to [0, \infty]$ is lower semicontinuous. In particular, for any $M > 0$, $\bracs{[\cdot]_{\text{var}, \rho} \le M} \subset E^{[a, b]}$ is closed. 
         \item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\text{var}, \rho}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $(E, \norm{\cdot}_E)$ is a normed vector space, then
     \begin{enumerate}
         \item[(5)] $f$ has at most countably many discontinuities.
@@ -87,7 +88,8 @@
     \begin{enumerate}
         \item[(a)] $|E_k| \ge N - k$.
         \item[(b)] $E_k \subset I_k^o$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     for $k = 1$.
 
     Let $k \le N$ and suppose inductively that $E_k, I_k$ have been constructed. Let $x_k \in E_k$, then by (b), there exists $\eps > 0$ such that $[x_k - \eps, x_k + \eps] \subset I_k$ and $|E_k \setminus [x_k - \eps, x_k + \eps]| \ge N - k$. Let $y_k \in [x_k - \eps, x_k + \eps]$ such that $\norm{f(x_k) - f(y_k)} \ge 1/n$, $I_{k + 1} = I_k \setminus [x_k - \eps, x_k + \eps]$, and $E_{k+1} = E_k \setminus [x_k - \eps, x_k + \eps]$, then $I_k$ and $E_k$ satisfies (a) and (b).

src/fa/rs/rs-bv.tex

@@ -35,7 +35,8 @@
     \begin{enumerate}
         \item[(a)] For each continuous seminorm $\rho$ on $E$, $[f_\alpha - f]_{u, \rho} \to 0$.
         \item[(b)] $\lim_{\alpha \in A}\int_a^b f_\alpha dG$ exists.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $f \in RS([a, b], G)$ and $\int_a^b f dG = \lim_{\alpha \in A}\int_a^b f_\alpha dG$. In particular,
     \begin{enumerate}
         \item If $H$ is complete, then condition (b) may be omitted.
@@ -55,11 +56,13 @@
     \begin{enumerate}
         \item $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$.
         \item $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$,
     \begin{enumerate}
         \item[(3)] $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG} < \eps/3$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Thus for any $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$,
     \[
     \rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps

src/fa/tvs/complete-metric.tex

@@ -21,12 +21,14 @@
     \begin{enumerate}
         \item[(a)] $\eta(y - Tx) \le \gamma \eta(y)$.
         \item[(b)] $\rho(x) \le C \eta(y)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that:
     \begin{enumerate}
         \item $\sum_{n \in \natp}\rho(x_n) \le C\eta(y)/(1 - \gamma)$.
         \item $y = \limv{N}\sum_{n = 1}^N Tx_n$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In particular,
     \[
     T\braks{B_E\paren{0, \frac{Cr}{(1 - \gamma)}}} \supset B_F(0, r)
@@ -38,12 +40,14 @@
     \begin{enumerate}
         \item[(I)] $\sum_{n = 1}^N\rho(x_n) \le C\eta(y)\sum_{n = 0}^{N-1}\gamma^{n}$.
         \item[(II)] $\eta\paren{y - \sum_{n = 1}^N Tx_n} \le \eta(y)\gamma^N$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By assumption, there exists $x_{N+1} \in E$ such that:
     \begin{enumerate}
         \item[(i)] $\eta\paren{y - \sum_{n = 1}^{N+1} Tx_n} \le \gamma \eta\paren{y - \sum_{n = 1}^N Tx_n} \le \gamma^{N+1}$.
         \item[(ii)] $\rho(x_{N+1}) \le C\eta\paren{y - \sum_{n = 1}^N Tx_n} \le C\eta(y)\gamma^N$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Combining (I) and (ii) shows that $\sum_{n = 1}^N \rho(x_n) \le C \eta(y) \sum_{n = 0}^N \gamma^n$. Therefore there exists $\seq{x_n} \subset E$ such that (I) and (II) holds for all $N \in \natp$.
 
     By (I), $\sum_{n \in \natp}\rho(x_n) \le C\eta(y)\sum_{n \in \natz}\gamma^n = C \eta(y)/(1 - \gamma)$. By (II), $\limv{N}\eta\paren{y - \limv{N}\sum_{n = 1}^N Tx_n} = \limv{N}\eta(y)\gamma^N = 0$.
@@ -55,7 +59,8 @@
     \begin{enumerate}
         \item[(a)] For any $r > 0$, there exists $\delta(r) > 0$ such that $\overline{T(B_E(0, r))} \supset B_F(0, \delta(r))$.
         \item[(b)] $E$ is complete.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for every $s > r$, $T(B_E(0, s)) \supset B_F(0, \delta(r))$.
 \end{proposition}
 \begin{proof}
@@ -65,12 +70,14 @@
         \item[(ii)] $s_1 = r$.
         \item[(iii)] For all $n \in \natp$, $\overline{T(B_E(0, s_n))} \supset B_F(0, \delta_n)$.
         \item[(iv)] $\rho_1 = \rho$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Let $y_0 \in B(0, r)$ and $x_0 = 0$. Let $N \in \natp$ and suppose inductively that $\bracs{x_n}_1^N \subset E$ has been constructed such that:
     \begin{enumerate}
         \item[(I)] For each $0 \le n \le N - 1$, $\rho(x_{n+1} - x_n) < s_n$.
         \item[(II)] For each $0 \le n \le N$, $\eta(Tx_n - y) \le \rho_{n+1}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By density of $T(x_N + B_E(0, s_N))$ in $Tx_N + B_F(0, \rho_N)$, there exists $x_{N+1} \in T(x_N + B_E(0, s_N))$ such that $\eta(Tx_{N+1} - y) \le \rho_{N+2}$.
 
     By (I), $\seq{x_N}$ is a Cauchy sequence, so
@@ -88,7 +95,8 @@
     \begin{enumerate}
         \item[(a)] For any $r > 0$, there exists $C \ge 0$ such that for any $y \in T(E)$, there exits $x \in T^{-1}(y)$ with $\rho(x) \le C\eta(y)$.
         \item[(b)] $E$ is complete.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then $T(E)$ is closed.
 \end{proposition}
 \begin{proof}

src/fa/tvs/completion.tex

@@ -8,11 +8,13 @@
         \item $\wh E$ is a complete separated TVS.
         \item $\iota \in L(E; \wh E)$.
         \item[(U)] For any $(F, T)$ satisfying (1) and (2), there exists a unique $\ol{T} \in L(\wh E; F)$ such that the following diagram commutes:
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(4)] $\iota(E)$ is dense in $\wh E$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(\wh E, \iota)$ is the \textbf{Hausdorff completion} of $E$.
 \end{definition}
 \begin{proof}

src/fa/tvs/continuous.tex

@@ -9,7 +9,8 @@
         \item $T \in UC(E; F)$.
         \item $T \in C(E; F)$.
         \item $T$ is continuous at $0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $T$ is a \textbf{continuous linear map}. The set $L(E; F)$ denotes the vector space of all continuous linear maps from $E$ to $F$.
 \end{definition}
 \begin{proof}
@@ -48,7 +49,8 @@
         }
         \]
 
-    \end{enumerate}
+    \end\{enumerate\}
+
     The uniformity $\fU$ and its induced topology are the \textbf{product uniformity/topology}, and $E$ equipped with $\fU$ is the \textbf{product TVS} of $\seqi{E}$.
 \end{definition}
 

src/fa/tvs/definition.tex

@@ -8,7 +8,8 @@
     \begin{enumerate}
         \item[(TVS1)] $E \times E \to E$ with $(x, y) \mapsto x + y$ is continuous.
         \item[(TVS2)] $K \times E \to E$ with $(\lambda, x) \mapsto \lambda x$ is continuous.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then the pair $(E, \topo)$ is a \textbf{topological vector space}.
 \end{definition}
 
@@ -50,7 +51,8 @@
     \begin{enumerate}
         \item There exists a unique translation-invariant uniformity $\fU$ on $E$ that induces the topology on $E$.
         \item For each neighbourhood $V \in \cn(0)$, let $U_V = \bracs{(x, y) \in E^2| x - y \in V}$, then for any fundamental system of neighbourhoods $\fB_0$ at $0$, $\fB = \bracs{U_V| V \in \fB_0}$ is a fundamental system of entourages for $\fU$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The space $E$ will always be assumed to be equipped with its translation-invariant uniformity.
 \end{proposition}
 \begin{proof}
@@ -162,12 +164,14 @@
     \begin{enumerate}
         \item[(TVB1)] For each $U \in \fB$, there exists $V \in \fB$ such that $V + V \subset U$.
         \item[(TVB2)] For each $U \in \fB$, $U$ is circled and radial.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $\fB \subset 2^E$ is a family of sets that contain $0$ and satisfies (TVB1) and (TVB2), then there exists a unique topology $\topo$ on $E$ such that:
     \begin{enumerate}
         \item $\topo$ is translation-invariant.
         \item $\fB$ is a fundamental system of neighbourhoods at $0$ for $\topo$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(3)] $(E, \topo)$ is a TVS.
@@ -186,7 +190,8 @@
         \item[(FB1)] For any $V, V' \in \fB$, there exists $W \in \fB$ with $W \subset V \cap V'$. In which case, $U_{V} \cap U_{V'} \supset U_W \in \mathfrak{V}$.
         \item[(UB1)] For any $x \in E$ and $V \in \fB$, $x - x = 0 \in V$, so $\Delta \subset U_V$.
         \item[(UB2)] For any $V \in \fB$, by (TVB1), there exists $W \in \fB$ such that $W + W \subset V$. In which case, for any $x, y, z \in E$ with $x - y, y - z \in W$, $x - z \in V$. Therefore $U_W \circ U_W \subset U_V$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\fU$ on $E$ for which $\mathfrak{V}$ is a fundamental system of entourages.
 
     (1): Since $\mathfrak{V}$ is translation-invariant, so is $\fU$.

src/fa/tvs/dual.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item $u \in \hom(E; \real)$ when $E$ is viewed as a vector space over $\real$.
         \item For any $x \in E$, $\dpb{x, \phi}{E} = \dpb{x, u}{E} - i \dpb{ix, u}{E}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Conversely, if $u \in \hom(E; \real)$ and $\phi \in \hom(E; \complex)$ is defined by $\dpb{x, \phi}{E} = \dpb{x, u}{E} - i \dpb{ix, u}{E}$ for all $x \in E$, then $f \in \hom(E; \complex)$.
 \end{proposition}
 \begin{proof}[Proof {{\cite[Proposition 5.5]{Folland}}}. ]

src/fa/tvs/inductive.tex

@@ -9,7 +9,8 @@
         \item For each $i \in I$, $T_i \in L(E_i; E)$.
         \item[(U)] For any topology $\mathcal{S}$ on $E$ satisfying (1) and (2), $\mathcal{S} \subset T$.
         \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T_i \in L(E_i; F)$ for all $i \in I$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The topology $\topo$ is the \textbf{inductive topology} on $E$ induced by $\seqi{T}$.
 \end{definition}
 \begin{proof}
@@ -61,7 +62,8 @@
 
         for all $i \in I$.
         \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T^i_E \in L(E_i; F)$ for all $i \in I$.
-    \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}

src/fa/tvs/metric.tex

@@ -82,7 +82,8 @@
     \begin{enumerate}
         \item[(a)] For each $n \in \natp$, $U_n$ is circled, radial, and contains $0$.
         \item[(b)] For each $n \in \natp$, $U_{n+1} + U_{n+1} \subset U_n$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a pseudonorm $\rho: E \to [0, \infty)$ such that for each $n \in \natp$,
     \[
     U_{n+1} \subset \rho^{-1}([0, 2^{-n})) \subset U_{n}
@@ -110,7 +111,8 @@
 
         so $\rho(\lambda x) \le \rho(x)$.
         \item[(PN3)] Let $x, y \in X$ and $M, N \subset \natp$ finite such that $x \in U_M$ and $y \in U_N$. Assume without loss of generality that $\rho_M + \rho_N < 1$, then there exists a unique $P \subset \nat$ finite such that $\rho_P = \rho_M + \rho_N$. In which case, $U_P \supset U_M + U_N$ by assumption (b). Therefore $\rho(x + y) \le \rho(x) + \rho(y)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     For any $x \in U_{n+1}$, $\rho(x) \le 2^{-n+1} < 2^n$, so $U_{n+1} \subset \rho^{-1}([0, 2^{-n}))$ by \autoref{proposition:dyadic-semigroup-order}. On the other hand, for any $x \in E$ with $\rho(x) < 2^{-n}$, $x \in U_{2^{-n}} = U_n$. This allows showing the remaining seminorm axioms by considering neighbourhoods of the form $\bracs{U_n|n \in \natp}$.
     \begin{enumerate}
         \item[(PN4)] Let $x \in X$ and $n \in \natp$. By assumption (a), there exists $\alpha > 0$ such that for any $\lambda \in K$ with $\abs{\lambda} \ge \alpha$, $x \in \lambda U_n$. Therefore for any $\lambda \in K$ with $\abs{\lambda} \le \alpha^{-1}$, $\lambda x \in U_n$, and $\rho(x) \le 2^{-n}$.

src/fa/tvs/projective.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item For each $i \in I$, $T_i \in L(E; F_i)$.
         \item[(U)] If $\mathfrak{V}$ is a uniformity on $E$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Moreover,
     \begin{enumerate}
         \item[(3)] $\fU$ is translation-invariant.
@@ -19,7 +20,8 @@
         \]
 
         is a fundamental system of neighbourhoods for $E$ at $0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The uniformity $\fU$ and its topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$.
 \end{definition}
@@ -74,7 +76,8 @@
 
         for all $i \in I$.
         \item For any TVS $F$ over $K$ and $S \in \hom(F; E)$, $S \in L(F; E)$ if and only if $T^E_i \circ S \in L(F; E_i)$ for all $i \in I$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the \textbf{projective limit} of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$.
 \end{definition}
 \begin{proof}

src/fa/tvs/quotient.tex

@@ -17,7 +17,8 @@
         \]
 
         If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$.
 \end{definition}
 \begin{proof}
@@ -32,7 +33,8 @@
     \begin{enumerate}
         \item[(TVB1)] Let $U \in \cn(0)$ be circled and radial. For any $\lambda \in K$ with $\abs{\lambda} \le 1$, $\lambda \pi(U) = \pi(\lambda U) \subset \pi(U)$, so $\pi(U)$ is also circled. For any $x + M \in E/M$, there exists $\lambda \in K$ such that $x \in \lambda U$. In which case, $x \in \lambda U + M = \pi(U)$, so $\pi(U)$ is also radial.
         \item[(TVB2)] For any $U \in \cn(0)$ circled and radial, by \autoref{proposition:tvs-good-neighbourhood-base}, there exists $W \in \cn(0)$ such that $W + W \subset U$. In which case, $\pi(W) + \pi(W) \subset \pi(U)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     By \autoref{proposition:tvs-0-neighbourhood-base}, there exists a unique translation-invariant topology on $E/M$ such that $\fB$ is a fundamental system of neighbourhoods at $0$, which must be the quotient topology on $E/M$. In which case, the quotient topology is a vector space topology by (3) of \autoref{proposition:tvs-0-neighbourhood-base}.
 
     (2), (3), (U): By \autoref{definition:quotient-topology}.

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

@@ -115,7 +115,8 @@
         \]
 
         is an isomorphism.
-    \end{enumerate}
+    \end\{enumerate\}
+
     which allows the identification
     \[
     \underbrace{B_{\sigma}(E; B_{\sigma}(E; \cdots)))}_{k \text{ times}} = B^k_{\sigma}(E; F)
@@ -166,7 +167,8 @@
     \begin{enumerate}
         \item[(a)] There exists a dense subset $S \subset E$ such that $T_\alpha x \to Tx$ strongly for all $x \in S$.
         \item[(b)] $\bracs{T_\alpha|\alpha \in A}$ is uniformly equicontinuous. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     then $T_\alpha \to T$ in $L_s(E; F)$.
 \end{proposition}