Weak Convergence implies boundedness and componentwise convergence Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in \ell^2$. 
Would you help me to prove that $\{\|  a_n\|\}$ bounded and for a fixed  $i$, $a_i^{(n)} \to a^{(n)}$ where $x^{(n)}$ denote the $n$-component of $x$. 

To show that a weakly convergent sequence is norm-bounded, usually the uniform boundedness principle is applied, see also this link. The convergence in the $n$-the component follows immediately, if you choose $y$ as the $n$-th unit vector in $l^2$.
