Definition 5.2.1 (Semivariation).label Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, $f:[a,b]\to E$ and $P\in\scr{P}([a,b])$ be a partition, then
\begin{align*}V_{\rho,P}(f)=\sums{n}{j=1}\rho(f(x_{j})-f(x_{j-1}))\end{align*}
is the variation of $f$ with respect to $\rho$ and $P$. The supremum over all such parititions
\begin{align*}V_{\rho}(f)=\SUP{P\in\scr{P}([a,b])}V_{\rho,P}(f)\end{align*}
is the semivariation of $f$ on $[a,b]$ with respect to $\rho$. If $E$ is a normed vector space, then the variation and semivariation of $f$ is taken with respect to its norm and denoted $V_{P}(f),V(f)$ respectively.
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