Characterizations of Vector-valued Weakly Almost Periodic Functions

We characterize the weak almost periodicity of a vector-valued, bounded, continuous function. We show that if the range of the function is relatively weakly compact, then the relative weak compactness of its right orbit is equivalent to that of its left orbit. At the same time, we give the function some other equivalent properties. 1. Introduction. Let S be a semitopological semigroup, let ᐄ be a Banach space, and let Ꮿ(S, ᐄ) be the space of bounded continuous functions from S to ᐄ with supremum norm. Let f ∈ Ꮿ(S, ᐄ). The right (left) translate of f by s ∈ S is the function R S f (L S f) such that R S f (t) = f (ts) and (L S f (t) = f (st))