Show that $\psi:L(E,L(E,F))\rightarrow L^2(E,F)$ given by $[\psi(T)](u,v)=[T(u)](v)$ is a linear homeomorphism

Let $E,F$ be Banach spaces. Denote by $L^2(E,F)$ the space of continuous bilinear applications $T:E^2\rightarrow F$ with norm $||T||=\underset {||x_i||_E≤1}{sup}{||T(x_1,x_2)||_F}$.

Show that $\psi:L(E,L(E,F))\rightarrow L^2(E,F)$ given by $[\psi(T)](u,v)=[T(u)](v)$ is a linear homeomorphism, first showing that $L^2(E,F)$ is Banach and then using the open mapping theorem.