Adjusted formulation of a total variation statement.

src/fa/lc/barrel.tex

@@ -23,7 +23,7 @@
     $(3) \Rightarrow (2)$: Let $D \subset E$ be a barrel and $\rho: E \to [0, \infty)$ be its gauge. By (4) of \autoref{definition:gauge}, $D = \bracs{\rho \le 1}$, so $\bracs{\rho > 1}$ is open, and $\rho$ is semicontinuous. By assumption, $\rho$ is continuous, so $D \in \cn_E(0)$ by (5) of \autoref{lemma:continuous-seminorm}. 
 \end{proof}
 
-\begin{summary}[Barreled Spaces]
+\begin{summary}
 \label{summary:barreled-space}
     The following types of locally convex spaces are barreled:
     \begin{enumerate}
@@ -61,7 +61,7 @@
 \label{proposition:barrel-limit}
     Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barreled.  
 \end{proposition}
-\begin{proof}[Proof, {{\cite[II.7.2]{SchaeferWolff}}}]
+\begin{proof}[Proof, {{\cite[II.7.2]{SchaeferWolff}}}. ]
     Let $D \subset E$ be a barrel, then for each $i \in I$, $T_i^{-1}(D) \subset E_i$ is also a barrel, and thus a neighbourhood of $0$ in $E_i$. By (5) of \autoref{definition:lc-inductive}, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barreled. 
 \end{proof}