Added the fundamental theorem of calculus.

src/fa/rs/bv.tex

@@ -6,7 +6,7 @@
 \label{definition:total-variation}
     Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, $f: [a, b] \to E$, and $P \in \scp([a, b])$ be a partition, then
     \[
-    V_{\rho, p}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
+    V_{\rho, P}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
     \]
 
     is the \textbf{variation} of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
@@ -19,7 +19,45 @@
     If $E$ is a normed vector space, then the variation and total variation of $f$ is taken with respect to its norm.
 \end{definition}
 
-\begin{definition}[Bounded Variation, {{\cite[Proposition X.1.1]{Lang}}}]
+\begin{definition}[Variation Function]
+\label{definition:variation-function}
+    Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, $f: [a, b] \to E$, then the function
+    \[
+    T_{f, \rho}(x) = \sup_{P \in \scp([a, x])}V_{\rho, P}(f) = [f|_{[a, x]}]_{\text{var}, \rho}
+    \]
+
+    is the \textbf{variation function} of $f$ with respect to $\rho$, and:
+    \begin{enumerate}
+        \item $T_{f, \rho}: [a, b] \to [0, \infty]$ is a non-negative, non-decreasing function. 
+        \item If $f \in BV([a, b]; E)$, then for any $[c, d] \subset [a, b]$, $[f]_{\text{var}, \rho} = T_{f, \rho}(d) - T_{f, \rho}(a)$. 
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    (2): Let $P \in \scp([a, c])$ and $Q = \seqf{x_j} \in \scp([a, d])$ be partitions containing $P$, then
+    \[
+    V_{\rho, Q}(f) - V_{\rho, P}(f) = \sum_{x_j > c}\rho(f(x_j) - f(x_{j - 1})) \le [f]_{\text{var}, \rho}
+    \]
+
+    As this holds for all $Q \in \scp([a, d])$ containing $P$, 
+    \[
+    T_{f, \rho}(d) - T_{f, \rho}(c) \le T_{f, \rho}(d) - V_{\rho, P}(f) \le [f|_{[c, d]}]_{\text{var}, \rho}
+    \]
+
+    On the other hand, for any $R \in \scp([c, d])$, $P \cup R \in \scp([a, d])$ and contains $P$. Therefore
+    \[
+    T_{f, \rho}(d) - V_{\rho, P}(f) \ge V_{\rho, R \cup P}(f) - V_{\rho, P}(f) = V_{\rho, R}(f)
+    \]
+
+    Since this holds for all $P \in \scp([a, c])$, 
+    \[
+    T_{f, \rho}(d) - T_{f, \rho}(c) \ge V_{\rho, R}(f)
+    \]
+
+    and as the above holds for all $R \in \scp([c, d])$, $T_{f, \rho}(d) - T_{f, \rho}(c) \ge [f|_{[c, d]}]_{\text{var}, \rho}$. 
+\end{proof}
+
+
+\begin{definition}[Bounded Variation]
 \label{definition:bounded-variation}
     Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, and $f: [a, b] \to E$. If $[f]_{\text{var}, \rho} < \infty$, then $f$ is of \textbf{bounded variation} with respect to $\rho$.
 
@@ -40,7 +78,7 @@
         \item[(5)] $f$ has at most countably many discontinuities.
     \end{enumerate}
 \end{definition}
-\begin{proof}
+\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),
     \[
     V_{\rho, P}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
@@ -71,7 +109,19 @@
 
 \begin{proposition}
 \label{proposition:bounded-variation-one-side-limit}
-    
+    Let $E$ be a complete locally convex space and $f \in BV([a, b]; E)$, then for each $x \in [a, b]$, the limits $\lim_{y \downto x}f(y)$ and $\lim_{y \upto x}f(y)$ exist. 
 \end{proposition}
+\begin{proof}
+    By flipping $f$, it is sufficient to consider the right-side limit $\lim_{y \downto x}f(y)$. 
+    
+    Let $\rho: E \to [0, \infty)$ be a continuous seminorm on $E$, and $T_{\rho, f}: [a, b] \to [0, \infty)$ be the variation function of $f$ with respect to $\rho$. For any $\eps > 0$, there exists $\delta > 0$ such that $T_{\rho, f}(z) - \lim_{y \downto x}T_{\rho, f}(y) < \eps$ for all $z \in (x, x + \delta)$. In which case, for any $x < y < z < x + \delta$, 
+    \[
+    \rho(f(z) - f(y)) \le [f|_{y, z}]_{\text{var}, \rho} \le T_{\rho, f}(z) - T_{\rho, f}(y) \le T_{\rho, f}(z) - \lim_{u \downto x}T_{\rho, f}(u) < \eps
+    \]
+
+    By completeness of $E$, the limit $\lim_{y \downto x}f(y)$ exists. 
+\end{proof}
+
+