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.