an:01618693
Zbl 0995.46025
Azagra, Daniel; Jim??nez-Sevilla, Mar
The failure of Rolle's theorem in infinite-dimensional Banach spaces
EN
J. Funct. Anal. 182, No. 1, 207-226 (2001).
00075195
2001
j
46G05 47H10
Rolle theorem; smooth norm; Brouwer fixed point theorem; bump
A bump is nonzero real function with bounded support. The authors prove that if a Banach space \(X\) admits a \(C^p\) smooth (Lipschitz) bump then it admits another \(C^p\) smooth (Lipschitz) bump \(f:X\to [0,1]\) with the property that \(f'(x)\neq 0\) for all x in the interior of the support of \(f\). This is applied to discussing Rolle's theorem, deleting diffeomorphisms, and Brouwer fixed points in infinite dimensions.
S.S.Kutateladze (Novosibirsk)