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The set of all e-periods for f is denoted by J ( f e). We say that f is almost periodic, written a.p., if for every e >0, the set J(J~e) is relatively dense in ,I1. A subset V of $ is called relatively dense (in $) if there exists an l > 0 such that every subinterval of $ of length I meets V. We call f weakly almost periodic, written w.a.p., if for each x' in the conjugate space X', the function x' of, mapping $ into the complex plane ~, is almost periodic. Continuous a.p. and continuous w.a.p, functions are treated in [1]. Let ~ be a family of functions mapping $ into X. We say that ~ is uniformly