Weakly almost periodic function

Remarks

Using right translates instead of left translates gives the same class of functions. Grothendieck's double limit criterion gives an equivalent form. The function $f$ is weakly almost periodic if and only if, for all sequences $(x_n)$ and $(y_m)$ in $G$, $$\lim_{n} \lim_{m} f(x_n y_m) = \lim_{m} \lim_{n} f(x_n y_m)$$ whenever both iterated limits exist. Every almost periodic function is weakly almost periodic, since norm-compact sets are weakly compact.

Example: on $G = \Z$, let $f = \mathbf{1}_{\set{0}}$. Its translates are the functions $\mathbf{1}_{\set{k}}$, which lie in the closed subspace $c_0(\Z)$. Every bounded linear functional on $c_0(\Z)$ is $h \mapsto \sum_n h(n) c_n$ for some $(c_n)$ with $\sum \abs{c_n} < \infty$, so $\mathbf{1}_{\set{k}} \to 0$ weakly as $\abs{k} \to \infty$. Hence every sequence of translates has a weakly convergent subsequence, with limit a translate or $0$, and by the Eberlein–Šmulian theorem $f$ is weakly almost periodic.

Non-example: on $G = \Z$, let $f = \mathbf{1}_{\set{0, 1, 2, \ldots}}$. Take $x_n = n$ and $y_m = -m$. Then $\lim_m f(n - m) = 0$ for each $n$, while $\lim_n f(n - m) = 1$ for each $m$. The iterated limits are $0$ and $1$, so $f$ is not weakly almost periodic.

On a locally compact group, every continuous function vanishing at infinity and every coefficient $x \mapsto \ip{\pi(x) u}{v}$ of a unitary representation is weakly almost periodic.