Let $H=\ell_2$ be the Hilbert space of the square-summable sequences where $ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i, \quad \|x\|=\sqrt{\langle x,x\rangle}. $ Let $F: H\rightarrow H$ be an affine mapping, i.e. $ F[\lambda u+(1-\lambda)v]=\lambda F(u)+(1-\lambda)F(v), \quad \forall u,v\in H, \lambda\in \mathbb{R}. $ Let $\{u^k\}$ be a sequence given by $ u^{k+1}=F(u^k) \quad k\in\mathbb{N}, $ where $u^0$ is an any point in $H$. Find the conditions on $F$ and $u^0$ such that $\{u^k\}$ is weakly convergent but not strongly convergent.
Example. If $u^0=(1,0,0,\ldots,0,\ldots)$ and $F(u)$ is given by $ F(u)=(0,u_1, u_2, \ldots, u_n, \ldots) \quad \forall u=(u_1,u_2,\ldots, u_n, \ldots)\in H. $ The sequence $\{u^k\}$ generated by the formula $u^{k+1}=F(u^k)$ is given by $ u^0=(1,0,\ldots, 0,\ldots), \quad u^1=(0,1,0,\ldots, 0, \ldots), \ldots, u^n=(0,0,\ldots, 1, 0, \ldots),\ldots $ is weakly convergent but not strongly convergent to $0\in H$.