Definition 13.5.2 (Path Integral).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, and $\gamma \in C([a, b]; F)$ be a path. For any $f: \gamma([a, b]) \to E$, $f$ is path-integrable with respect to $\gamma$ if $f \circ \gamma \in RS([a, b], \gamma; E)$. In which case,
\[\int_{\gamma} f = \int_{a}^{b} f(\gamma(t)) \gamma(dt)\]
is the path integral of $f$ with respect to $\gamma$. The set $PI([a, b], \gamma; E)$ is the space of all functions path-integrable with respect to $\gamma$.