Let $E$ be a Banach space, $p \in (1, \infty)$, and $L_p := L_{p}(X, \mu, E)$. Is $(L_p)^* \cong L_{p'}$ where $\frac{1}{p} + \frac{1}{p'} = 1$?

Question

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use Bochner integral. The space $L_p := L_{p}(X, \mu, E)$ consists of (equivalence classes of) all Bochner measurable functions $f$ with values in $E$ whose norm $|f|$ lies in the standard $L_{p}(X, \mu, \mathbb R_{\ge 0})$ space.

A standard result is that if $E = \mathbb R$ and $p \in (1, \infty)$ then the continuous dual space $(L_p)^*$ of $L_p$ is indeed $L_{p'}$ where $\frac{1}{p} + \frac{1}{p'} = 1$. One of the proof relies on the continuous linear operator $T:L_{p'} \to (L_p)^*$ defined by $$ \langle T u, f\rangle := \int_X u(x)f(x)\mathrm d \mu(x) \quad \forall u \in L_{p'}, \forall f \in L_{p}. $$

Then $T$ is indeed a linear surjective isometry. However, I could not see how to use the same approach for a general Banach space $E$.

Is there such a linear surjective isometry $T$ in case $E$ is a general Banach space?

Answer 1

As @geetha290krm answered in a comment, the reference is Vector Measures by Joseph Diestel, John Jerry Uhl. In particular, Theorem 1 at page 98 is exactly what we are looking for.

THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leqq p<\infty$, and $X$ be a Banach space. Then $L_{p}(\mu, X)^{*}=L_{q}\left(\mu, X^{*}\right)$ where $p^{-1}+q^{-1}=1$, if and only if $X^{*}$ has the Radon-Nikodým property with respect to $\mu$.