Definition 11.2.1 (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
\[V_{\rho, p}(f) = \sum_{j = 1}^{n} \rho(f(x_{j}) - f(x_{j - 1}))\]
is the variation of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
\[[f]_{\text{var}, \rho}= \sup_{P \in \scp([a, b])}V_{\rho, P}(f)\]
is the total variation of $f$ on $[a, b]$ with respect to $\rho$.
If $E$ is a normed space, then the variation and total variation of $f$ is taken with respect to its norm.