Define by putting:
Let be range of ,
Then we have:
(1)
(2) There exists a continuous functional such that and for all (This functional is called a Banach limit)
I would like to know how to show:
(3) for all
(4) when converge
Let If for all then for all Therefore Since it follows that So which implies that
For (4), I think you need to show that every element of with only finitely many nonzero coordinates is in the range of It follows that is unchanged if finitely many coordinates of are changed. Now suppose that as Given we can assume (by changing finitely many coordinates of ) that for all n. It will then follow from (3) that Now let to conclude that