Local dual spaces of Banach spaces of vector-valued functions

We show that $L_\infty(\mu,X^*)$ is a local dual of $L_1(\mu,X)$, and $L_1(\mu,X^*)$ is a local dual of $L_\infty(\mu,X)$, where $X$ is a Banach space. A local dual space of a Banach space $Y$ is a subspace $Z$ of $Y^*$ so that we have a local representation of $Y^*$ in $Z$ satisfying the properties of the representation of $X^{**}$ in $X$ provided by the principle of local reflexivity.