By symmetry, it is sufficient to determine the unit ball in the sector . Let f be a real-valued differentiable function on an interval I . By the Mean Value Theorem, there exists a point such that as claimed. Conversely, if f is Lipschitz continuous with some constant K , as well as differentiable, then Taking the supremum over , we see that any Lipschitz constant for f cannot be smaller than L . Remark Note that f does not satisfy the assumptions of the Contraction Mapping Principle, since it s not a strict contraction - its optimal Lipschitz constant is .
Theorem Let and be the spaces of all sequences of real numbers such that respectively, is finite.
We will find the following lemma useful in the proof that and are complete. To see completeness of , let be a Cauchy sequence in . We want to show that the sequence converges to a limit in . By definition of the norm, the i -th coordinates again form a Cauchy sequence in . Then where we have used the Lemma in the last step.