Adjusted labeling of conditions for Banach-Steinhaus.

src/fa/norm/normed.tex

@@ -73,8 +73,8 @@
 \label{theorem:uniform-boundedness}
     Let $E, F$ be normed vector spaces and $\mathcal{T} \subset L(E; F)$. If
     \begin{enumerate}
-        \item[(B)] $E$ is a Banach space.
+        \item[(B1)] $E$ is a Banach space.
-        \item[(E2)] For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$.
+        \item[(B2)] For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$.
     \end{enumerate}
 
     then $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)} < \infty$.

src/fa/tvs/equicontinuous.tex

@@ -37,13 +37,13 @@
 \label{theorem:banach-steinhaus}
     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[(B1)] $E$ is a Baire space. 
-        \item[(B')] $E$ is barrelled and $F$ is locally convex.
+        \item[(B1')] $E$ is barrelled and $F$ is locally convex.
     \end{enumerate}
 
     and that
     \begin{enumerate}
-        \item[(E2)] For each $x \in E$, $\alg(x) = \bracs{Tx|T \in \alg}$ is bounded in $F$. 
+        \item[(B2')] For each $x \in E$, $\alg(x) = \bracs{Tx|T \in \alg}$ is bounded in $F$. 
     \end{enumerate}
 
     then
@@ -54,14 +54,14 @@
     \end{enumerate}
 \end{theorem}
 \begin{proof}[Proof, {{\cite[IV.4.2]{SchaeferWolff}}}. ]
-    (B) + (E2) $\Rightarrow$ (E1): Let $V \in \cn_F(0)$ be closed and circled, then $U = \bigcap_{T \in \alg}T^{-1}(V)$ is circled and closed. By (E2), $U$ is absorbing, so $E = \bigcup_{n \in \natp}nU$. Since $E$ is Baire, there exists $n \in \natp$, $W \in \cn_E(0)$, and $x \in E$ such that $x + W \subset nU$. As $U$ is circled,
+    (B1) + (B2) $\Rightarrow$ (E1): Let $V \in \cn_F(0)$ be closed and circled, then $U = \bigcap_{T \in \alg}T^{-1}(V)$ is circled and closed. By (B2), $U$ is absorbing, so $E = \bigcup_{n \in \natp}nU$. Since $E$ is Baire, there exists $n \in \natp$, $W \in \cn_E(0)$, and $x \in E$ such that $x + W \subset nU$. As $U$ is circled,
     \[
     W \subset nU - nU = nU + nU = 2nU
     \]
 
     so $U \in \cn_E(0)$, and $\alg$ is equicontinuous by \autoref{proposition:equicontinuous-linear}. 
 
-    (B') + (E2) $\Rightarrow$ (E1): Let $V \in \cn_F(0)$ be convex, circled, and closed, then $U = \bigcap_{T \in \alg}T^{-1}(V)$ is convex, circled, and closed. By (E2), $U$ is absorbing, and hence a barrel in $E$. By (B'), $U \in \cn_E(0)$, $\alg$ is equicontinuous by \autoref{proposition:equicontinuous-linear}. 
+    (B1') + (B2) $\Rightarrow$ (E1): Let $V \in \cn_F(0)$ be convex, circled, and closed, then $U = \bigcap_{T \in \alg}T^{-1}(V)$ is convex, circled, and closed. By (B2), $U$ is absorbing, and hence a barrel in $E$. By (B1'), $U \in \cn_E(0)$, $\alg$ is equicontinuous by \autoref{proposition:equicontinuous-linear}. 
 
     (E1) $\Rightarrow$ (C1) + (C2): By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} and \autoref{proposition:equicontinuous-linear-closure}.
 \end{proof}