This is my first topic here, please apologise if I am doing something wrong.
Let $X$ and $Y$ be Banach spaces with their duals $X^*$ and $Y^*$, respectively, and $F:X\to Y$ with the Frechet gradient Lipschitz continuous with constant $L$. Let $z\in Y^*$ with $\|z\|=1$ (can be also $\leq 1$ or equal to some given constant). What can be said about the Lipschitz continuity of the gradient of the function defined by $x \mapsto \langle z , F(x) \rangle$? One has $∇ \langle z, F(x) \rangle = z \circ ∇F(x)$, but from here I am lost and I need to show that $∇ \langle z, F(x) \rangle$ is Lipschitz continuous and to determine the corresponding Lipschitz constant.
Thank you all in advance for any hint or answer, which will be credited in my Master' Thesis.