I am interested in determining whether the function $F:L^{\infty}(0,\infty;L^{\infty}(0,1)) \to \mathbb R$ defined by $$u \mapsto F(u)=\int_0^\infty \int_0^1 u(x,t)^2 \ \mathrm dx \, \mathrm dt $$ is differentiable/Frechét differentiable at an element $u \in L^2(0,\infty;L^2(0,1))$.

Any suggestions or assistance would be greatly appreciated.