Definition 5.2.3 (Variation Function).label Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$ and $f:[a,b]\to E$, then the function
\begin{align*}T_{f,\rho}(x)=\SUP{P\in\scr{P}([a,x])}V_{\rho,P}(f)=V_{\rho}(f|_{[a,x]})\end{align*}
is the variation function of $f$ with respect to $\rho$, and:
- (1)
$T_{f,\rho}:[a,b]\to [0,\infty]$ is a non-negative, non-decreasing function
- (2)
If $f\in BV([a,b];E)$, then for any $[c,d]\suf [a,b]$, $V_{\rho}(f|_{[c,d]})=T_{f,\rho}(d)-T_{f,\rho}(c)$
Proof. Let $P\in\scr{P}([a,c])$ and $Q=\curl{x_j}^{n}_{j=0}\in \scr{P}([a,d])$ be partitions containing $P$ then
\begin{align*}V_{\rho,Q}(f)-V_{\rho,P}=\sums{}{x_j>c}\rho(f(x_{j})-f(x_{j-1}))\leq V_{\rho}(f)\end{align*}
Taking supremum over $Q$ and $P$ respectively we have
\begin{align*}T_{f,\rho}(d)-T_{f,\rho}(c)\leq T_{f,\rho}(d)-V_{\rho,P}(f)\leq V_{\rho}(f|_{[c,d]})\end{align*}
OTOH for any $R\in\scr{P}([c,d])$, $P\cup R\in\scr{P}([a,d])$ and contains $P$ hence
\begin{align*}T_{f,\rho}(d)-V_{\rho,P}(f)\geq V_{\rho,P\cup R}(f)-V_{\rho,P}(f)=V_{\rho,R}(f)\end{align*}
Taking supremum over $P$ and $R$ respectively we obtain
\begin{align*}T_{f,\rho}(d)-T_{f,\rho}(c)\geq T_{f,\rho}(d)-V_{\rho,P}(f)\geq V_{\rho}(f|_{[c,d]})\end{align*}
$\square$