Let $F: X \to Y$ be a continuously differentiable function between banach spaces $X$ and $Y$. (we also can assume $X$ is reflexive space) Let $x_0 \in X$, this tells us the derivative $\nabla F(x_0) \in B(X, Y)$ is a bounded linear operator, this is also true for adjoint-derivative i.e., $\nabla F(x_0)^* \in B(Y^* , X^*)$. My question is that if we have convergent nets $x_i \to x_0$ and $y_i^* \overset{w^*}{\rightarrow} y^*$ in $X$ and $Y^*$ respectively then do we have $$ \nabla F(x_i)^* y_{i}^* \overset{w^*}{\rightarrow} \nabla F(x_0)^* y^* \in X^* $$
My thought: I think $ G(x) = \nabla F(x)^*$ in norm to norm, continuous as $H(x) = \nabla F(x) $ is norm to norm continuous because $F$ is continuously Frechet differentiable. Then we can get the result from Uniform boundedness.