Uniform boundedness theorems for nearly additive mappings

We deal with mappings from a (not necessarily commutative) groupG into a Banach spaceY which are nearly additive in the sense of satisfying that for some constantK ≥ 0, wheneverx _i andy _i ∈ G are such that Σ _i=1 ⁿ x _i = Σ _j=1 ^m y _j ,where P ’ is a fixed (non-negative) ”control” functional onG. Such maps, called zero-additive, appear in various contexts. The smallest constantK for which the inequality holds shall be noted byZ(F). For mappingsG’ Y we consider the (possibly infinite) distance Then one may ask whether or not a zero-additive mapF must be near to a true additive mapA : G → Y in the sense of dist(F, A) < ∞ and howZ(F) and dist(F, A) are related (a question which goes back to Ulam). We prove the following “uniform boundedness” result, thus solving a problem stated by CASTILLO and the present author. This work is supported in part by DGICYT project PB97-0377 and HI project 1997-0016.