For each $n$, let $\{A_j^n\}_j$ be a disjoint partition of the interval $[0,1]$ such that as $n$ increases, the partition gets finer.
Suppose that we have $$f_n(t) = \sum_{j=1}^n a_{jn}\chi_{A_j^n}(t)$$ where $a_{jn} \in V$ belongs to a Hilbert space $V \subset\subset H$. We have that $f_n$ is bounded uniformly in $L^\infty(0,T;V)$ so there is a subsequence that converges weak-* in that space.
Is there anyway to extract pointwise a.e. in $t$ convergence for $f_n$ (or a subsequence) or any sort of strong convergence in Bochner space? I thought that since $f_n$ is defined everywhere and is continuous a.e., and indeed is piecewise constant in time, we might gains something.