A note on spaces of almost periodic functions with values in Banach spaces

Abstract In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach X . context, it is known that normality or Bochner-type property, which characterizes these functions, based relative compactness family translates. Now, prove every class sequentially compact and translates function belonging to subspace dense its own class, i.e., condition periodicity $f\in AP(\mathbb yields sequence f has subsequence converges equivalent This extends previous work by same authors case numerical functions.