Re: functions of bounded variation in Banach spaces
- From: Ronald Bruck <bruck@xxxxxxxxxxxxxxxxx>
- Date: Mon, 15 Aug 2005 16:34:34 -0700
In article
<32786093.1124137363341.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Michael <www.maimonid@xxxxxxxxxxx> wrote:
> Hello,
>
> I have desperately sought the answer of the following
> question: Let E is a Banach space and f:[a,b]-->E be of bounded variation
> (that is, sup \Sum_p ||f(v)-f(u)||< oo , where the suppremum is taken over
> all the subdivision p={[u,v]} of [a,b]into non-overlapping intervals [u,v]).
> Is f differentiable almost everywhere?
> if E=R, this is of course the famous theorem of Lebesgue. But I ask in the
> general case. If anyone knows about this
> question, please let me know.
If E is reflexive, yes. If E is not reflexive, no.
IIRC, a proof can be found in Brezis' little book on monotone
operators. If not there, try Barbu's book on nonlinear semigroups.
It's a pretty standard result (but I don't have either of those books
here to check).
--Ron Bruck
.
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