Almost periodic functions and representations in locally convex spaces

Properties of diverse classes of almost periodic functions with values in locally convex spaces and of almost periodic representations on locally convex spaces are considered. The well-known criterion for the almost periodicity of weakly almost periodic group representations on Banach spaces (in terms of scalar almost periodicity) is extended to the case of weakly continuous weakly almost periodic representations on barrelled spaces in which the weakly closed convex hulls of weakly compact sets are weakly compact. Applications of this result are indicated and a survey of the current state of some other classical problems in the theory of almost periodic functions (as applied to almost periodic functions with values in locally convex spaces) and modern directions of investigation related to almost periodic functions on groups and finite-dimensional unitary representations of groups are presented. In particular, decomposition problems for weakly almost periodic representations and characterizations of diverse classes of almost periodic functions (including criteria for almost periodicity), existence problems for the mean value, countability conditions for the spectrum of a scalarly almost periodic function, theorems on the integral and the differences of almost periodic functions, and other relationships among strong, scalar, and weak almost periodicity for functions with values in locally convex spaces are treated.