Dual space of $l^p(\mathbb{Z},X)$

Question

Let $X$ be a Banach space, $p \in [1,\infty)$ and $l^p(\mathbb{Z},X)$ the usual sequence space taking values in $X$. Is it always true that $(l^p(\mathbb{Z},X))^* = l^q(\mathbb{Z},X^*)$ and $(c_0(\mathbb{Z},X))^* = l^1(\mathbb{Z},X^*)$, where $\frac{1}{p} + \frac{1}{q} = 1$? I do know that for a $\sigma$-finite measure $\mu$ it holds $(L^p(\mu,X))^* = L^q(\mu,X^*)$ if and only if $X^*$ has the Radon-Nikodym property w.r.t. $\mu$. I suppose that the Radon-Nikodym property is trivially satisfied here, but I am not entirely sure. As I am not an expert in this field, a short proof or a reference would be appreciated (or a counterexample if I am wrong, of course). Thanks!