Necessary condition for a minimizer in a convex and closed set

Question

I want to prove the following result

Let $K\subset H$ a convex and closed set. Let $J$ be a functional over $K$ that is continuous and differentiable at the point $u$ which is a minimizer. Then $J’(u)(v-u)\geq 0$ $\forall v\in K$

H is a Hilbert space.

Here is my attempt :

Consider $v\in K$ ($v\neq u$) and $\theta\in (0,1)$. We know that $J(u)\leq J(u + \theta(v-u))$. This leads to

$$ J(u)\leq J(u) + \theta J’(u)(v-u) + o(\theta(v-u)) $$

Thus

$$ 0\leq J’(u)(v-u) + \frac{o(\theta(v-u))}{\theta\lVert v-u\rVert}\lVert v-u\rVert $$

And we conclude by taking the limit as $\theta$ goes to $0^{+}$.

I am wondering if this is correct because I did not use the fact that $K$ is closed.

Thank you a lot

Answer 1

Yes, your proof is correct; closedness is not needed. It is typically added "automatically", because it is often crucial to proof existence of minimizers.

By the way: in your "result" the assumption that $u$ is a minimizer is missing.