Assume we have $N \in C^1(\mathbb{R}^3)$ which is bounded and has bounded gradient. Let $X := L^2([0, T])$ (for some $T>0$) and consider the operator $T: X \times X \rightarrow \mathcal{L}(X \times X, X)$ given by $$ T:(\alpha, \beta) \mapsto \Big{(} (\varphi, \psi) \mapsto \partial_1 \big{(}N(\alpha(\tau), \beta(\tau), \tau) \cdot\varphi(\tau)\big{)} + \partial_2 \big{(}N(\alpha(\tau), \beta(\tau), \tau)\cdot \psi(\tau)\big{)} \Big{)}. $$
I wonder if this is continuous. $X$ is equipped with the usual norm and $X \times $X with the usual product topology. The norm on $\mathcal{L}(X \times X, X)$ is the operator norm.
I indeed managed to show, that if $(\alpha_k, \beta_k) \rightarrow (\alpha, \beta)$, then $$ \lVert T(\alpha_k, \beta_k)(\varphi, \psi)- T(\alpha, \beta)(\varphi, \psi) \rVert_{X} \rightarrow 0 $$ for fixed $(\varphi, \psi) \in X \times X$ with $\lVert (\varphi, \psi) \rVert_{X \times X} = 1$. This is of course not quite enough for convergence in operator norm...
Can someone help me out?