non convergent sequence under continuous linear functional on $l_2$

non convergent sequence under continuous linear functional on $l_2$

Does the sequence $\{e_i\}$ converges in $l_2$? does the sequence
$\{f(e_i)\}$ converges for any continuous linear functional on $l_2$?
I know that in $l_2$ norm $\{e_i\}$ does not converges as
$d(l_i,l_j)=\sqrt{2}$ always! But I dont know the other! could any one
help me?