Let $\Omega$ be a bounded domain of $\mathbb{R}^N$ and $1<p<+\infty$. Consider the Sobolev space
$$W^{1,p}(\Omega) = \{ u \in L^p(\Omega) : |\nabla u| \in L^p(\Omega)\}$$
Let us denote by $W'$ the dual of $W^{1,p}(\Omega)$. There is a well-known characterization for $W'$. If $T \in W'$, then there exist $g_1 \in [L^{p'}(\Omega)]^N$ and $g_2 \in L^{p'}(\Omega)$ such that
$$ T(u) = \int_{\Omega} \vec{g_1} \cdot \nabla u + g_2 u dx, \quad \forall u \in W^{1,p}(\Omega) $$
However, I would like to know if these functions $g_1$ and $g_2$ are unique (a.e). In demonstrating this result, this is not clear, as these functions are obtained by extending a certain continuous linear functional using the Hahn-Banach Theorem, however, as in general, this theorem does not guarantee uniqueness of the extension, we could choose another extension and thus achieve other functions $\vec{g_3} \in [L^{p'}(\Omega)]^N$ and $g_4 \in L^{p'}(\Omega)$ in order to represent $T$.
One more question: Suppose that i have proved that a certain functional $T$ belongs to $W'$ and that i can write
$$ T(u) = \int_{\Omega} \vec{f_1} \cdot \nabla u + f_2 u dx $$
Can I then conclude that $\vec{f_1} \in [L^{p'}(\Omega)]^N$ and that $f_2 \in L^{p'}(\Omega)$?
Any help or reference would be greatly appreciated