Updated spelling of barreled to barrelled for consistency.

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