At each point/function f0 for all functions $u$, $\nabla F$ should satisfy

$$\langle \langle \nabla F (f_0), u \rangle \rangle = D_u F = \lim_{\epsilon \to 0} \frac{F(f_0 + \epsilon u) - F(f_0)}{\epsilon}$$

$$= \lim_{\epsilon \to 0} \frac{\langle \langle f_0 + \epsilon u, g \rangle \rangle - \langle \langle f_0 , g \rangle \rangle}{\epsilon} $$

$$= \lim_{\epsilon \to 0} \frac{\langle \langle \epsilon u, g \rangle \rangle}{\epsilon} $$

$$ = \lim_{\epsilon \to 0} \langle \langle u, g \rangle \rangle$$

$$= \langle \langle u, g \rangle \rangle$$

The only solution to this $\langle \langle \nabla F (f_0), u \rangle \rangle = \langle \langle u, g \rangle \rangle$ for all $u$ is to let

$$\nabla F(f_0) = g$$