Infinite Dimensional Version of Rolle′s Theorem
Yang Cai-ping · Journal of Civil Aviation University of China · 2010
Let Ω be a bounded open convex subset of a reflexive real Banach space X and let Y be a real normed linear space.We obtain an infinite dimensional version of Rolle′s theorem:if the operator A∶■→Y is strongly continuous on ■,Frechet differentiable in Ω and there exists a non-zero continuous linear functional f of Y such that(fAx)= 0 for all x∈Ω,then there exists at least one point ∈Ω such that f(A(′)u)= 0 for all u∈X.