Let $(X,\|\cdot\|)$ be a (infinite dimensional) Banach space and $f_{n}:[0,1]\longrightarrow (X,\|\cdot\|)$ continuous, for each integer $n\geq 1$, a sequence of mappings. There is some "pointwise compactness criteria" for this sequence? That is, a criteria which state (of course, under suitable conditions) the convergence of some subsequence of the sequence $f_{n}(t)$, for a given $t\in [0,1]$.
Many thanks in advance for your comments.