I want to understand every step of the proof found on wikipedia.
It says that we let ${\kappa _{q}} :L^{p}(\mu ) \rightarrow L^{q}(\mu )^{*}$ be the isometry between $L^p$ and it's dual (this is given by the Riesz representation theorem) and then apply the following composition with the transpose (adjoint) of ${\kappa _{q}}$.
$j_{p}:L^{p}(\mu ){\overset {\kappa _{q}}{\longrightarrow }}L^{q}(\mu )^{*}{\overset {\left(\kappa _{p}^{-1}\right)^{*}}{\longrightarrow }}L^{p}(\mu )^{**}$
I am trying to understand why the adjoint is a bijective map. I have constructed it as follows:
$${\left(\kappa _{p}^{-1}\right)^{*}}( F) (\phi) = \phi \circ \kappa _{p}^{-1}(F)$$
where $\phi \in L^{q}(\mu )^{*}$, $F \in L^{q}(\mu )^{*}$. When $\phi$ is fixed and $F$ varies $\left(\kappa _{p}^{-1}\right)^{*}(F) : L^{q}(\mu )^{*} \rightarrow R$. When $\phi$ varies we have
$$ {\left(\kappa _{p}^{-1}\right)^{*}} (\phi) = \phi \circ \kappa _{p}^{-1} $$
So $\left(\kappa _{p}^{-1}\right)^{*} : L^{q}(\mu )^{*} \rightarrow L^{q}(\mu )^{**}$.
(Have I understood correctly up to now? also feel free to correct my terminology)
Now If want to show that $\left(\kappa _{p}^{-1}\right)^{*}$ is bijective. I know that $\kappa _{p}^{-1}$ is by the Riesz Representation theorem so I only need to check $\phi$, but $\phi$ is my input so it will always be bijective and thus I am done.
Have I understood correctly the proof? I feel like I might be wrong.
Now I am only left to show that this map coincides with the canonical embedding $J$ of $L^p$ into its bidual.
this is because $$ \left( \kappa _{p}^{-1}\right)^{*} \circ \kappa _{p} = \phi \circ \kappa _{p}^{-1} \circ \kappa _{p} = \phi $$
and $J(\phi)(f) = \phi(f)$
Where $f \in L^p$, is the canonical embedding.