无穷维情形下的Rolle定理

Infinite Dimensional Version of Rolle′s Theorem

设Ω是自反的实Banach空间X中的有界开凸集,Y为一实赋范线性空间。证明了一个无穷维情形下的Rolle定理:如果算子A∶■→Y在■上强连续,在Ω内Frèchet可微,并且存在Y上的非0连续线性泛函f,使得f(Ax)=0对一切x∈Ω成立,则至少存在一点∈Ω,使对一切u∈X,都成立f(A′()u)=0。

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 ■,Frèchet 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.