Let be a real Banach space with the dual . Fix and call a linear functional almost norming for if and . In Part 1 I showed that in any Banach space there exists a continuous selection of almost norming functionals. Here I will prove that there is no uniformly continuous selection in .
Claim. Let be the unit sphere in , the -dimensional -space. Suppose that is a map such that is almost norming in the above sense. Then the modulus of continuity satisfies .
(If an uniformly continuous selection was available in , it would yield selections in with a modulus of continuity independent of .)
Proof. Write . For any we have , hence
for all . Sum over all and change the order of summation:
There exists such that
Fix this from now on. Define to be the same vector as , but with the th component flipped. Rewrite the previous sum as
and add them together:
Since , it follows that , as claimed.