I have an operator $T:H \to O(H,H)$ where $H$ is a Hilbert space and by $O(H,H)$ I mean the space of operators $F$ with $F:H \to H$. We are in the general nonlinear setting.
What exactly would it mean to say that $T$ is continuous? If $h_n \to h$ in $H$, $T(h_n) \to T(h)$ should hold in which norm?