Fenchel-Rockafellar duality: From Banach space to Hilbert space

Let $\mathbb{B}$, $\mathbb{H}$ be respectively a Banach, Hilbert space. Let $f: \mathbb{H} \longrightarrow \mathbb{R}$, $g: \mathbb{B} \longrightarrow \mathbb{R}$ and $A: \mathbb{B} \longrightarrow \mathbb{H}$. Consider the following ubiquitous primal problem
$$\min_{x\in \mathbb{B}} \hspace{0.5cm} f(Ax)+g(x)$$
It is well-known that the Fenchel-Rockafellar dual problem is
$$\max_{u\in \mathbb{H}} \hspace{0.5cm} -f^*(-u)-g^*(A^*u)$$
where $f^*: \mathbb{H} \longrightarrow \mathbb{R}$, $g^*: \mathbb{B^*} \longrightarrow \mathbb{R}$ and $A^*: \mathbb{H} \longrightarrow \mathbb{B^*}$. Here $f^*, g^*$ are convex conjugate of $f$ and $g$, respectively, and $A^*$ is the adjoint operator of $A$.

In this post, we would like to show this result by using the change of variable.
Let $v=Ax$, the primal problem can be rewritten, assuming that $A$ is surjective, i.e. $\text{Im}A=\mathbb{H}$,
$$\min_{v\in \mathbb{H}} \hspace{0.5cm} f(v)+\hat{g}(v)$$
where $\hat{g}(v):= \inf \{g(x): Ax=v\}$
By applying Fenchel duality problem, we obtain the following dual problem
$$\max_{x\in \mathbb{B}} \hspace{0.5cm} -f^*(-u)-\hat{g}^*(u)$$
The rest is to show that
$$\hat{g}^*(u)=g^*(A^*u)$$

Indeed,
$$\hat{g}^*(u)=\sup_{v} \langle u, v\rangle - \hat{g}(u)= \sup_{x} \langle u, Ax\rangle - g(x)= g^*(A^*u)$$
We obtained desired result.

With this observation, we see that the second primal problem only involves the Hilbert space, which is more easier for theoretical analysis, for example, the optimality conditions.