If $\{u_n\}$ is bounded in a real Hilbert space $H$, with inner product $(\cdot,\cdot)$, then ${\|u_n\|^2u_n}$ is also bounded.

As there is a weakly converging sub-sequence, we can WLOG assume that $\{\|u_n\|^2u_n\}$ converges weakly to $u_0\in H$.

Is it right that $u_0=\|u\|^2u$ ? Practically speaking, can we repeatedly choose a sub-sequence of $\{u_n\}$ to obtain $（\|u_n\|^2u_n，v)\rightarrow (\|u\|^2u,v), \, \forall v\in H $?