Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$.
Let $u_n \longrightarrow u$ strongly in $L^2(0,T;L^p(\Omega))$ for $1 \leq p < \infty$.
It is known that strong convergence in $L^2$ implies a.e. convergence along a subsequence.
Can I then imply that $u_{n_k}(\cdot, t) \longrightarrow u(\cdot,t) $ strongly in $L^p(\Omega)$ for a.e. $t \in \left[0,T\right]$?
Or are there any problems since I work in a Bochner space?