Updated spelling of barreled to barrelled for consistency.

src/fa/lc/barrel.tex

@@ -25,14 +25,14 @@
 
 \begin{summary}
 \label{summary:barreled-space}
-    The following types of locally convex spaces are barreled:
+    The following types of locally convex spaces are barrelled:
     \begin{enumerate}
         \item Every locally convex space with the Baire property.
         \item Every Banach space and every Fréchet space. 
-        \item Inductive limits of barreled spaces. 
+        \item Inductive limits of barrelled spaces. 
         \item Spaces of type (LB) and (LF). 
-        \item The locally convex direct sum of barreled spaces. 
-        \item Products of barreled spaces.
+        \item The locally convex direct sum of barrelled spaces. 
+        \item Products of barrelled spaces.
     \end{enumerate}
 \end{summary}
 \begin{proof}
@@ -46,7 +46,7 @@
 
 \begin{proposition}
 \label{proposition:baire-barrel}
-    Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barreled. 
+    Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barrelled. 
 \end{proposition}
 \begin{proof}[Proof, {{\cite[II.7.1]{SchaeferWolff}}}. ]
     Let $D \subset E$ be a Barrel, then $E = \bigcup_{n \in \natp}nD$ is a countable union of closed sets. Since $E$ is Baire, there exists $n \in \natp$, $U \in \cn_E(0)$ circled, and $x \in E$ such that $x + U \in nB$. In which case, 
@@ -59,9 +59,9 @@
 
 \begin{proposition}
 \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.  
+    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 barrelled.  
 \end{proposition}
 \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. 
+    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 barrelled. 
 \end{proof}
 

src/fa/tvs/equicontinuous.tex

@@ -16,6 +16,15 @@
     (5) $\Rightarrow$ (1): Let $V \in \cn_F(0)$, then $U = \bigcap_{T \in \alg}T^{-1}(V) \in \cn_E(0)$. Thus for any $x, y \in E$ with $x - y \in U$, $Tx - Ty \in V$ for all $T \in \alg$. 
 \end{proof}
 
+\begin{proposition}
+\label{proposition:equicontinuous-bounded}
+    Let $E, F$ be TVSs over $K \in \RC$ and $\alg \subset L(E; F)$ be equicontinuous, then for any ideal $\sigma \subset \mathfrak{B}(E)$, $\alg$ is a bounded subset of $B_\sigma(E; F)$. 
+\end{proposition}
+\begin{proof}
+    Let $S \in \sigma$ and $U \in \cn_F(0)$, then there exists $V \in \cn_E(0)$ such that $\bigcup_{T \in \alg}T(V) \subset U$. Since $S$ is bounded, there exists $\lambda \in K$ such that $S \subset \lambda V$. Therefore $\bigcup_{T \in \alg}T(S) \subset \lambda U$, and $\alg$ is bounded in $B_\sigma(E; F)$. 
+\end{proof}
+
+
 \begin{proposition}[{{\cite[IV.4.3]{SchaeferWolff}}}]
 \label{proposition:equicontinuous-linear-closure}
     Let $E, F$ be TVSs over $K \in \RC$ and $\alg \subset L(E; F)$ be equicontinuous, and $\alg'$ be the closure of $\alg$ in $F^E$ with respect to the product topology, then $\alg'$ is equicontinuous and hence $\alg' \subset L(E; F)$. 
@@ -29,7 +38,7 @@
     Let $E, F$ be TVSs over $K \in \RC$ and $\alg \subset L(E; F)$. Suppose that one of the following holds:
     \begin{enumerate}
         \item[(B)] $E$ is a Baire space. 
-        \item[(B')] $E$ is barreled and $F$ is locally convex.
+        \item[(B')] $E$ is barrelled and $F$ is locally convex.
     \end{enumerate}
 
     and that
@@ -86,7 +95,7 @@
     Let $E, F, G$ be TVSs over $K \in \RC$ and $\alg$ be separately continuous bilinear maps from $E \times F$ to $G$. If one of the following holds:
     \begin{enumerate}
         \item[(B)] $E$ is Baire.
-        \item[(B')] $E$ is barreled and $G$ is locally convex. 
+        \item[(B')] $E$ is barrelled and $G$ is locally convex. 
     \end{enumerate}
 
     and that