DIFFERENTIABLE FUNCTIONS ON CERTAIN ... - Project Euclid
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DIFFERENTIABLE FUNCTIONS ON CERTAIN ... - Project Euclid
erally, if F is any Banach space and Æ : CID-+F is completely con- tinuous and differentiable in D, then Æ(CLD) C CI/(dD). Note that these results are false if C(0, ...
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Communicated by G. A. Hunt, October 21, 1964
The main result in this note, Theorem 2, can be thought of as a very strong maximum modulus type theorem. For example, let D be a bounded connected open set in C(0, 1), and l e t / : C\D—>Rn be continuous and differentiate in D. Then ƒ is determined by its values on the boundary of D. More exactly, ƒ(CLD)CCl/(d£>). More generally, if F is any Banach space and ƒ : CID-+F is completely continuous and differentiable in D, then ƒ(CLD) C CI/(dD). Note that these results are false if C(0, 1) is replaced by a Hubert space. 1. Let D be a connected bounded open set in lp where p is not an even integer. Assume f is a real-valued function, continuous on C\D and n-times differentiable in D with n^p. Then f(CW)
