I have a question which maybe looks very simple: Let $T$ be an orthogonal projection on a Hilbert space $H$. If $g(x,u)\in H$, for all $u\in \mathbb R$, and the inner product is defined by $\langle f(.), g(.,u)\rangle_{H}=\int_{\mathbb R}f(x)g(x,u)dx $ which is a function of $u$ (say $h(u)$), for all $f\in H$.
Now my question is that if we apply the projection to the resulting function $h(u)$, can we move the projection inside the integral, i.e.:
$T (h(u))= T\big( \int_{\mathbb R}f(x)g(x,u)dx \big)= \int_{\mathbb R}f(x)T(g(x,u))dx$
(If this is not always true what are the cases where we can do this?)