Let $\phi : E \to F$ be a linear mapping with restriction $\phi_1: E_1$ $\to F_1.$ Suppose that $\phi^* : F^* \to E^*$ is dual to $\phi$.Then $\phi^*$ can be restricted to the pair $(F_1^\perp, E_1^\perp).$
I have proved that indeed $\phi^*$ can be restricted to the given pair.However, my question is that is $\phi_1$ dual to the induced map from the $\phi^*$.
For me, it should be because the necessary and sufficient condition for the maps to be dual map is satisfied by the original maps already.