Definition 20.2.3 (Bochner Integral). Let $(X, \cm)$ be a measurable space, $E, F$ be normed spaces over $K \in \RC$, $G$ be a Banach space over $K$,
\[\lambda: E \times F \to G \quad (x, y) \mapsto xy\]
be a bounded bilinear map, and $\mu: \cm \to F$ be a vector measure, then there exists a unique $I_{\lambda} \in L(L^{1}(X, |\mu|; E); G)$ such that:
For any $x \in E$ and $A \in \cm$, $I_{\lambda}(x \cdot \one_{A}) = x \mu(A)$.
For any $f \in L^{1}(X, |\mu|; E)$, $\normn{I_\lambda f}_{G}\le \norm{\lambda}_{L^2(E, F; G)}\cdot \norm{f}_{L^1(X, |\mu|; E)}$.
For any $f \in L^{1}(X; E)$, $I_{\lambda} f = \int \lambda(f, d\mu)$ is the Bochner integral of $f$ with respect to $\mu$ and $\lambda$.
Proof. Same as Definition 20.2.1.$\square$