Definition 13.2.2: Variation Function | Jerry's Digital Garden

Definition 13.2.2 (Variation Function).label 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 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 $[c, d] \subset [a, b]$, $[f]_{\text{var}, \rho}= T_{f, \rho}(d) - T_{f, \rho}(a)$.

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 $R \in \scp([c, d])$, $T_{f, \rho}(d) - T_{f, \rho}(c) \ge [f|_{[c, d]}]_{\text{var}, \rho}$.$\square$