If $L$ is closed then the preimage of every compact set is closed. In particular $\ker L$ is a closed subspace of $E$.
Suppose $C\in F$ is compact, then we know from topology that the projection $\Prn_E:E\times C\rar E$ sends closed sets to closed sets and thus $[L\in C]=\Prn(\G(L)\cap(E\times C))$ is closed.