I know in my last post I said that I would be focusing on the Prime Number Theorem, but I want to make a brief digression on a topic related to an earlier post, namely holomorphic functions taking values in a Banach space. A discussion about Stein interpolation in my analysis class today is the impetus behind this digression.
Let be a Banach space, and be a domain. Recall that a function is said to be differentiable at if there exists such that
In this post, I want to prove an equivalent characterization of holomorphicity that relates such a function to complex-differentiable functions as studied in a first course in complex analysis.
Theorem 1 A function is holomorphic if and only if for every bounded linear function , the function defined by is holomorphic.
Proof: We showed in the post on the generalization of Liouville’s theorem that the function is holomorphic with derivative if is holomorphic, which is the direction. We now prove . Let and let be a postively oriented circle centered at with radius . For each , the function is an analytic function of . For in the interior of , we have by the Cauchy integral formula applied to , and that
Under the canonical injection, we can identify . Since is continuous in and is compact, there exists such that for all . Thus, the family of linear operators is pointwise bounded. By the principle of uniform boundedness, we have
Hence if ,
Taking the supremum over all , we conclude that
which converges to as . This shows that the difference quotients are Cauchy. Since is complete, as .