Updated formulation in BV.

src/fa/rs/bv.tex

@@ -65,12 +65,7 @@
     \begin{enumerate}
         \item $BV([a, b]; E)$ is a vector space.
         \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$.
-        \item Let $\fF$ be a filter on $BV([a, b]; E)$ and $f: [a, b] \to E$. If
+        \item For each continuous seminorm $\rho$ on $E$ and $M > 0$, $\bracs{[\cdot]_{\text{var}, \rho} \le M} \subset E^{[a, b]}$ is closed. 
-        \begin{enumerate}[label=\alph*]
-            \item $\pi_x(\fF) \to f(x)$ for all $x \in [a, b]$.
-            \item For every continuous seminorm $\rho$ on $E$, there exists $U \in \fF$ such that $\sup_{g \in U}[g]_{\text{var}, \rho} = M_\rho < \infty$.
-        \end{enumerate}
-        then $f \in BV([a, b]; E)$ with $[f]_{\text{var}, \rho} \le M_\rho$.
         \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}
     If $(E, \norm{\cdot}_E)$ is a normed vector space, then
@@ -79,14 +74,15 @@
     \end{enumerate}
 \end{definition}
 \begin{proof}[Proof {{\cite[Proposition X.1.1]{Lang}}}. ]
-    (3): Let $\rho$ be a continuous seminorm on $E$ and $P \in \scp([a, b])$, then by assumption (a),
+    (3): Let $\rho$ be a continuous seminorm on $E$, $P = \seqf{x_j} \in \scp([a, b])$, and $f \in \overline{\bracs{[\cdot]_{\text{var}, \rho} \le M}}$. For any $\eps > 0$, there exists $g \in \bracs{[\cdot]_{\text{var}, \rho} \le M}$ such that $\rho(f(x_j) - g(x_j)) < \eps$ for each $1 \le j \le n$. In which case, 
-    \[
+    \begin{align*}
-    V_{\rho, P}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
+    V_{\rho, P}(f) &= \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1})) \\
-    = \lim_{g, \fF}\sum_{j = 1}^n \rho(g(x_j) - g(x_{j - 1}))
+    &\le 2n\eps + \sum_{j = 1}^n \rho(g(x_j) - g(x_{j - 1})) \\
-    = \lim_{g \in \fF}V_{\rho, P}(g)
+    &\le V_{\rho, P}(g) + 2n\eps \le M + 2n\eps
-    \]
+    \end{align*}
     
-    By assumption (b), $[0, M_\rho]$ is in the filter generated by $V_{\rho, P}(\fF)$. Thus $V_{\rho, P}(f) \le M_\rho$. As this holds for all $P \in \scp([a, b])$, $V_{\rho, P}(f) \le M_\rho$, and $f \in BV([a, b]; E)$.
+
+    As the above holds for all $\eps > 0$, $V_{\rho, P}(f) \le M$. Since this holds for all $P \in \scp([a, b])$, $[f]_{\text{var}, \rho} \le M$.
 
     (5): For each $n \in \nat^+$, let
     \[