Given a functional $E:Z\rightarrow \mathbb{R}$ defined as $$ E(z) = G(F(z)), $$ where
- $F: Z \rightarrow H$ is an operator between Banach spaces and
- $G: H \rightarrow \mathbb{R}$ is a functional,
the derivative of $E$ (in a Banach space context, provided the necessary hypothesis are satisfied) computed by using the chain rule is $$ E' = G'(F(z))\circ F'(z). $$ Now, how can we show the relation between the duality pairings, i.e. $$ \langle E'(z), h\rangle_{Z^*\times Z} = \langle G'(F(z)), F'(z)h\rangle_{H^*\times H}\;? $$ I know that the adjoint operators are used in some way but cannot figure out how.