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 **analytic**on 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 .

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