71 lines
4.0 KiB
TeX
71 lines
4.0 KiB
TeX
\section{Functions of Bounded Variation}
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\label{section:bv}
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\begin{definition}[Total Variation]
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\label{definition:total-variation}
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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
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\[
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V_{\rho, p}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
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\]
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is the \textbf{variation} of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
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\[
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[f]_{\text{var}, \rho} = \sup_{P \in \scp([a, b])}V_{\rho, P}(f)
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\]
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is the \textbf{total variation} of $f$ on $[a, b]$ with respect to $\rho$.
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If $E$ is a normed space, then the variation and total variation of $f$ is taken with respect to its norm.
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\end{definition}
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\begin{definition}[Bounded Variation, {{\cite[Proposition X.1.1]{Lang}}}]
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\label{definition:bounded-variation}
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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 \textbf{bounded variation} with respect to $\rho$.
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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
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\begin{enumerate}
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\item $BV([a, b]; E)$ is a vector space.
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\item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$.
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\item Let $\fF$ be a filter on $BV([a, b]; E)$ and $f: [a, b] \to E$. If
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\begin{enumerate}
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\item $\pi_x(\fF) \to f(x)$ for all $x \in [a, b]$.
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\item For every continuous seminorm $\rho$ on $E$, there exists $U \in \fF$ such that $\sup_{g \in U}[g]_{\text{var}, \rho} = M_\rho < \infty$.
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\end{enumerate}
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then $f \in BV([a, b]; E)$ with $[f]_{\text{var}, \rho} \le M_\rho$.
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\item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\text{var}, \rho}$.
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\end{enumerate}
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If $(E, \norm{\cdot}_E)$ is a normed space, then
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\begin{enumerate}
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\item[(5)] $f$ has at most countably many discontinuities.
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\end{enumerate}
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\end{definition}
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\begin{proof}
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(3): Let $\rho$ be a continuous seminorm on $E$ and $P \in \scp([a, b])$, then by assumption (a),
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\[
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V_{\rho, P}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
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= \lim_{g, \fF}\sum_{j = 1}^n \rho(g(x_j) - g(x_{j - 1}))
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= \lim_{g \in \fF}V_{\rho, P}(g)
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\]
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By assumption (b), $[0, M_\rho]$ is in the filter generated by $V_{\rho, P}(\fF)$. Thus $V_{\rho, P}(f) \le M_\rho$. As this holds for all $P \in \scp([a, b])$, $V_{\rho, P}(f) \le M_\rho$, and $f \in BV([a, b]; E)$.
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(5): For each $n \in \nat^+$, let
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\[
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D_n = \bracs{x \in [a, b]|\forall \eps > 0, \exists y \in (x - \eps, x + \eps): \norm{f(x) - f(y)}_E \ge 1/n}
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\]
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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.
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Fix $N \in \nat^+$. Let $E_1 = D_n \cap (a, b)$ and $I_1 = (a, b)$, then
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\begin{enumerate}
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\item[(a)] $|E_k| \ge N - k$.
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\item[(b)] $E_k \subset I_k^o$.
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\end{enumerate}
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for $k = 1$.
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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).
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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 $[f]_{\text{var}} = \infty$.
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\end{proof}
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