The following question had appeared in so many places, but none justify it, I tried a lot but . If someone can give me a hand.
Let $X$ be a Hilbert space and $I:X\rightarrow \mathbb{R}$ a functional. If $(u_n)\subset X$ is such that $u_n$ converges weakly to $u$ in $X$ and $I'(u_n)\varphi \rightarrow 0$ for every $\varphi \in X$, then $I'(u)=0$, where $I'$ denotes the derivative of $I$.
Observations: 1) we know that $\|I'(u_n)\|\rightarrow 0$ (this is a fact in all the cases where this problem happened, so it can be used).
2) we don't know nothing special about $I$ besides being of class $C^1$.
3) I tried to prove that I get convergence $I(u_n)\varphi \rightarrow I(u)\varphi$ on $\mathbb{R}$ so I could use weak lower semicontinuity (w.l.s.) of the module function. I also tried something about weak converge on $X^\ast$ so I could use w.l.s. of the dual norm, but once again I wasn't sucessful.
I appreciate any help. Thanks in advance.