13.2 Functions of Bounded Variation
Definition 13.2.1 (Total Variation).label 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
is the variation of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
is the total variation of $f$ on $[a, b]$ with respect to $\rho$.
If $E$ is a normed vector space, then the variation and total variation of $f$ is taken with respect to its norm.
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
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 $[c, d] \subset [a, b]$, $[f]_{\text{var}, \rho}= T_{f, \rho}(d) - T_{f, \rho}(c)$.
Proof. (2): Let $P \in \scp([a, c])$ and $Q = \seqf{x_j}\in \scp([a, d])$ be partitions containing $P$, then
As this holds for all $Q \in \scp([a, d])$ containing $P$,
On the other hand, for any $R \in \scp([c, d])$, $P \cup R \in \scp([a, d])$ and contains $P$. Therefore
Since this holds for all $P \in \scp([a, c])$,
and as the above holds for all $R \in \scp([c, d])$, $T_{f, \rho}(d) - T_{f, \rho}(c) \ge [f|_{[c, d]}]_{\text{var}, \rho}$.$\square$
Definition 13.2.3 (Bounded Variation).label 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 bounded variation with respect to $\rho$.
The space $BV([a, b]; E)$ is the set of functions $[a, b] \to E$ of bounded variation with respect to every continuous seminorm on $E$, and
- (1)
$BV([a, b]; E)$ is a vector space.
- (2)
For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$.
- (3)
For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}: E^{[a, b]}\to [0, \infty]$ is lower semicontinuous. In particular, for any $M > 0$, $\bracs{[\cdot]_{\text{var}, \rho} \le M}\subset E^{[a, b]}$ is closed.
- (4)
For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\text{var}, \rho}$.
If $(E, \norm{\cdot}_{E})$ is a normed vector space, then
- (5)
$f$ has at most countably many discontinuities.
Proof [Proposition X.1.1, Lan93]. (3): For each $P \in \scp([a, b])$, the mapping $V_{P, \rho}: E^{[a, b]}\to [0, \infty]$ is continuous. Since $[\cdot]_{\text{var}, \rho}= \sup_{P \in \scp([a, b])}V_{P, \rho}$, $[\cdot]_{\text{var}, \rho}$ is lower semicontinuous by Proposition 5.22.2.
(5): For each $n \in \nat^{+}$, let
then $D = \bigcup_{n \in \nat^+}D_{n}$ is the set of discontinuity points of $f$. If $D$ is uncountable, then there exists $N \in \nat^{+}$ such that $D_{n}$ is infinite.
Fix $N \in \nat^{+}$. Let $E_{1} = D_{n} \cap (a, b)$ and $I_{1} = (a, b)$, then
- (a)
$|E_{k}| \ge N - k$.
- (b)
$E_{k} \subset I_{k}^{o}$.
for $k = 1$.
Let $k \le N$ and suppose inductively that $E_{k}, I_{k}$ have been constructed. Let $x_{k} \in E_{k}$, then by (b), there exists $\eps > 0$ such that $[x_{k} - \eps, x_{k} + \eps] \subset I_{k}$ and $|E_{k} \setminus [x_{k} - \eps, x_{k} + \eps]| \ge N - k$. Let $y_{k} \in [x_{k} - \eps, x_{k} + \eps]$ such that $\norm{f(x_k) - f(y_k)}\ge 1/n$, $I_{k + 1}= I_{k} \setminus [x_{k} - \eps, x_{k} + \eps]$, and $E_{k+1}= E_{k} \setminus [x_{k} - \eps, x_{k} + \eps]$, then $I_{k}$ and $E_{k}$ satisfies (a) and (b).
Therefore there exists pairs $\bracs{(x_k, y_k)|1 \le k \le N}$ such that $\norm{f(x_k) - f(y_k)}_{E} \ge 1/n$ for all $n$, and the smallest interval containing each $(x_{k}, y_{k})$ are pairwise disjoint. Thus $[f]_{\text{var}}\ge N/n$ for all $N \in \nat^{+}$, so $[f]_{\text{var}}= \infty$.$\square$
Proposition 13.2.4.label 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.
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 $x < y < z < x + \delta$,
By completeness of $E$, the limit $\lim_{y \downto x}f(y)$ exists.$\square$
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