I know that $L_q \cong L^*_p$ (the two are isomorphic) if $\frac{1}{p} + \frac{1}{q} = 1$. Additionally, the map between the two $\phi : L_q \rightarrow L^*_p$ can be constructed as:
$$(\phi(g))(f) = F_g(f) = \int g(x)f(x)dx, g \in L_q, f \in L_p, F_g \in L^*_p$$
What would be the inverse map $\phi^{-1} : L^*_p \rightarrow L_q$ in that case? Is it a function $(\phi^{-1}(F)) \in L_q$ such that:
$$F(f) = \int (\phi^{-1}(F))(x)f(x)dx, \forall f\in L_p$$
Is this function $(\phi^{-1}(F))$ uniquely defined from the above?