Lemma 13.5.6.label Let $[a, b] \subset \real$, $E$ be a separated locally convex space over $K \in \RC$, $F$ be a Banach space over $K$, $H$ be a complete locally convex space over $K$, all 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 piecewise $C^{1}$ curve that is constant on $[a, a + \eps)$ and $(b - \eps, b]$, and $U \in \cn_{F}(\gamma([a, b]))$.
Extend $\gamma$ to $\real$ by
For each $\varphi \in C_{c}^{\infty}(\real; \real)$ with $\int_{\real} \varphi = 1$ and $t > 0$, let
then
- (1)
For each $t > 0$, $\gamma_{t} \in C^{\infty}([a, b]; F)$.
- (2)
There exists $t > 0$ such that for any $s \in (0, t)$, $\gamma_{s}(a) = \gamma(a)$ and $\gamma_{s}(b) = \gamma(b)$.
- (3)
For any $f \in C(U; E)$,
\[\int_{\gamma} f = \lim_{t \downto 0}\int_{\gamma_t}f\]
Proof. (1): By the Mean Value Theorem, for each $x, y \in [a, b]$,
By the Dominated Convergence Theorem, $\gamma_{t} \in C^{\infty}([a, b]; F)$.
(2): For sufficiently small $t$, $\supp{\varphi}\subset (-\eps, \eps)$. In which case, by assumption, $\gamma_{t}(a) = \gamma(a)$ and $\gamma_{t}(b) = \gamma(b)$.
(3): Since $\gamma$ is piecewise $C^{1}$ and $\gamma_{t} \in C^{\infty}([a, b]; F)$,
by the Dominated Convergence Theorem.$\square$
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