The classical version of Liouville’s theorem states that if is an entire bounded function, then is constant. One might ask if an analogous result holds for complex-differentiable functions taking values in a Banach space . The answer is yes.
\bigskip Let be a Banach space, and let be a domain (i.e., a connected open subset). Recall that a function is said to be analyticon if, for each , there exists such that
Lemma 1 If is an analytic function and is a continuous linear functional, then the function defined by
is analytic with derivative .
Proof:Fix a continuous linear functional and define as above. Since is bounded and linear, we have for and , that
In particular, we see that if is an entire function in the extended Banach-space-valued sense, then, for any , is an entire function in the classical sense. We are now ready to state and prove the generalization of Liouville’s theorem to -valued functions.
Theorem 2 Let be an entire function such that for all , for some finite . Then there exists such that for all . In other words, is constant.
Proof:Assume the contrary. Then there exists such that . By the Hahn-Banach theorem, there exists such that . Since
we see that is constant by the classical Liouville’s theorem. But this contradicts that .