Problem in functional analysis

Hello,

One defines .

is defined on the maximal subregion .

Now let and .

How to prove, that the following implication is **not** true in general:

In a book about functional analysis I read the following lemma:

"Let . If the inequality holds for all and for all then is weakly differentiable in with ."

Here the problem is , so the lemma can not be used.

Do you have a counter example?

Kind regards,

Alexander

Re: Problem in functional analysis

I think that you even have this in the classical sense - a function with directional derivatives at a point may not be differentiable there.