The Hahn-Banach Theorem on Arbitrary Groups

In this paper, one kind of subgroup in arbitrary group which similar to the linear subspace was constructed, and the generalization of the Hahn-Banach theorem on this kind of subgroup in arbitrary groups was obtained. The Hahn-Banach theorem is a powerful existence theorem whose form is par- ticularly appropriate to applications in linear problems. In its elegance and power, the Hahn-Banach theorem is a favorite of almost every analyst. The generalization of the Hahn-Banach theorem on groups has been discussed in many articles, much of these discussions were under the assumption of some condition of groups, such as the weakly commutativity in the paper of Z. Gajda and Z. Kominek (3), or groups in class G during R. Badora (1). The purpose of this paper is to find the sucient and necessary condition of Hahn-Banach theorem on arbitrary groups. Let G be a group, p,f be functionals on G ! R, then p is subadditive and f is additive if and only if p(xy) p(x) + p(y), x,y 2 G and f(xy) = f(x) + f(y), x,y 2 G. Moreover, p is completely commutative if and only if for any n 2 Z + and any per- mutation xk1,··· ,xkn of x1,··· ,xn 2 G, one has p( n Y i=1 xi) = p( n Y i=1 xki ).