Lemma 13.5.5.label Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $\gamma \in C([a, b]; F)$ be a rectifiable path, and $U \in \cn_{F}(\gamma([a, b]))$, then for any continuous seminorm $[\cdot]_{G}: G \to [0, \infty)$, $\eps > 0$, and $f \in C(U; E)$, there exists a piecewise linear path $\Gamma \in C([a, b]; F)$ such that:
- (1)
$\Gamma(a) = \gamma(a)$ and $\Gamma(b) = \gamma(b)$.
- (2)
$\braks{\int_\gamma f - \int_\Gamma f}_{F} < \epsilon$.
Proof. Let $[\cdot]_{E}: E \to [0, \infty)$ and $[\cdot]_{F}: F \to [0, \infty)$ such that for any $x \in E$ and $y \in F$, $[xy]_{G} \le [x]_{E}[y]_{F}$. Since $\gamma([a, b])$ is compact, by modifying $[\cdot]_{F}$, assume without loss of generality that there exists $V \in \cn_{F}(\gamma([a, b]))$ such that for any $x, y \in V$ with $[x - y]_{F} \le 1$, $[f(x) - f(y)]_{E} \le \eps$.
Since $f \in C(U; E)$, $f \in PI([a, b], \gamma; E)$ by Proposition 13.4.3. Given that $\gamma$ is of bounded variation, there exists $(P = \seqfz{x_j}, c) \in \scp_{t}([a, b])$ such that:
- (a)
For each $1 \le j \le n$,
\[\gamma([x_{j-1}, x_{j}]) \subset \bracs{y \in F|[y - x_{j-1}]_F \le 1}\] - (b)
$\braks{\int_\gamma f - S(P, c, f \circ \gamma, \gamma)}_{G} < \epsilon$.
Let $\Gamma$ be the piecewise linear path formed by linearing $f$ at points in $P$. For any $(Q, d) \in \scp_{t}([a, b])$ with $(Q, d) \ge (P, c)$,
As $\Gamma$ is also of bounded variation, $f \in PI([a, b], \Gamma; E)$. Since the above holds for all refinements of $(Q, d)$,
$\square$