Is $ G(x) = \nabla F(x)^* $ continuous when is $F$ continuously differentiable in Frechet's sense?

Question

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 the function $ G(x) = \nabla F(x)^* $ is norm to norm continuous ?

Answer 1

If $\|x_n-x\| \to 0$ the $\|\nabla F(x_n)-\nabla F(x)\|\to 0$ which implies $\|G(x_n)-G(x)\| \to 0$.