Editing Almost periodic function (section)

===Almost periodic functions on a locally compact group===
With these theoretical developments and the advent of abstract methods (the [[Peter&ndash;Weyl theorem]], [[Pontryagin duality]] and [[Banach algebra]]s) a general theory became possible. The general idea of almost-periodicity in relation to a [[locally compact abelian group]] ''G'' becomes that of a function ''F'' in ''L''<sup>∞</sup>(''G''), such that its translates by ''G'' form a [[relatively compact]] set.
Equivalently, the space of almost periodic functions is the norm closure of the finite linear combinations of characters of&nbsp;''G''. If ''G'' is compact the almost periodic functions are the same as the continuous functions.

The [[Bohr compactification]] of ''G'' is the  compact abelian group  of all possibly discontinuous characters of the dual group of ''G'', and is a compact group containing ''G'' as a dense subgroup. The space of uniform almost periodic functions on ''G'' can be identified with the space of all continuous functions on the Bohr compactification of&nbsp;''G''. More generally the Bohr compactification can be defined for any topological group&nbsp;''G'', and the spaces of continuous or ''L''<sup>''p''</sup> functions on the Bohr compactification can be considered as almost periodic functions on&nbsp;''G''.
For locally compact connected groups ''G'' the map from ''G'' to its Bohr compactification is injective if and only if ''G'' is a central extension of a compact group,  or equivalently the product of a compact group and a finite-dimensional vector space.

A function on a locally compact group is called ''weakly almost periodic'' if its orbit is weakly relatively compact in <math>L^\infty</math>.

Given a topological dynamical system <math>(X,G)</math> consisting of a compact topological space ''X'' with an action of the locally compact group ''G'', a continuous function on ''X'' is (weakly) almost periodic if its orbit is (weakly) precompact in the Banach space <math>C(X)</math>.