Let G be an almost metrizable topological group (for example, a locally compact group). This paper is concerned with the proof of two principal results. First, the following criterion for equicontinuity is proved: Let X be a union of G Ξ΄ -subsets of G, Y a uniform space and H a set of continuous mappings of X into Y ; then for H to be equicontinuous, it is sufficient (and obviously also necessary) that the sequence ((hn(x), hn(xn))) nβN be eventually in every entourage of Y for each sequence (hn) nβN in H, each x β X and each sequence (xn) nβN in X such that limnβ+β xn = x. Δoban's theorem on dyadicity is a basic tool in the proof of this result. Let e be the identity element of G; it follows immediately from the above criterion that G has equal left and right uniform structures if (and only if) limnβ+β anb -1 n = e for all sequences (an) nβN and (bn) nβN in G such that limnβ+β a -1 n bn = e (a well-known property in the case when G is metrizable). The second principal result is the following: Let us suppose that the left and right uniform structures on the almost metrizable topological group G are distinct; then the Banach space UR(G) contains a linear isometric copy L of l β such that inf{ 2l + h | h β U(G)} l for all l β L (and consequently, the quotient Banach space UR(G)/U(G) is nonseparable); moreover, if G is complete, then " " can be replaced by "=" (and consequently, UR(G)/U(G) contains a linear isometric copy of l β ).