# Banach limit

Banach limits originated in [a1], Chapt. II, Sect. 3. Denoting the positive integers by , the set is the real vector space of all bounded sequences of real numbers. For any element , one defines by for all . S. Banach showed that there exists an element in the dual , called , such that

1) for all ;

2) for all non-negative sequences ;

3) for all ;

4) for all convergent sequences . Banach proved the existence of this generalized limit by using the Hahn–Banach theorem. Today (1996), Banach limits are studied via the notion of amenability.

For a semi-group one defines to be the real vector space of all real bounded functions on . For an element one denotes the left (respectively, right) shift by (respectively, ). Thus, for all and for all . An element is called a left- (respectively right-) invariant mean if

1) ;

2) (respectively, ), where, e.g., is the adjoint of . itself is called left (respectively, right) amenable if there exists a left- (respectively, right-) invariant mean in . The existence of Banach limits above is a special case of an invariant mean, where equals the semi-group of natural numbers. Banach also proved that the real numbers are amenable (left and right). M.M. Day has proved that every Abelian semi-group is left and right amenable. On the other hand, , the free group on two generators, is not amenable.