Given a function $f$, let us define the translates $f_t(x)=f(x-t)$. A (Bochner) almost-periodic function is a bounded continuous function on $\mathbb R^\nu$ such that the set of functions $\{f_t\vert t\in\mathbb R^\nu\}$ form a precompact set with respect to the supremum norm (a precompact set is a set whose closure is compact).

This definition is taken from Appendix 1 of this paper.

I was wondering if we insisted that the set is compact rather than just precompact, is this equivalent to $f$ being periodic? That is,

Is it true that a bounded continuous function $f$ on $\mathbb R^\nu$ is periodic if and only if the set of functions $\{f_t\vert t\in\mathbb R^\nu\}$ form a compact set with respect to the supremum norm?