Weakly almost periodic function

Summary

A weakly almost periodic function on a topological group is a bounded continuous function with a special property. Its translates form a relatively compact set for the weak topology of the space of bounded continuous functions. The notion, due to Eberlein, weakens the uniform compactness that defines almost periodic functions. Every almost periodic function is weakly almost periodic, and on a locally compact group so is every continuous function vanishing at infinity. The weakly almost periodic functions carry a unique invariant mean, which makes them a natural setting for mean ergodic theorems.