The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces

详细信息    查看全文

• 作者：Azagra ; Daniel ; Jimé ; nez-Sevilla ; Mar
• 刊名：Journal of Functional Analysis
• 出版年：2001
• 出版时间：May 10, 2001
• 年：2001
• 卷：182
• 期：1
• 页码：207-226
• 全文大小：171 K

文摘

We prove the following new characterization of C^p (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C^p smooth (Lipschitz) bump function if and only if it has another C^p smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The “twisted tube” method we use in the proof is interesting in itself, as it provides other useful characterizations of C^p smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.