On this link http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst_2004;task=show_msg;msg=1414.0001 is the argument that a linear subspace in a normed space is closed w.r.t. norm iff it is weakly closed.
On the other hand, $c_0$ (sequences convergent to $0$) is a norm-closed linear subspace of $l_\infty$ (bounded sequences), but it is not weakly closed, since the base vectors $e_i$ are weakly dense in $l_\infty$.
Since I studied func.an. quite a while ago, my question is - what am I missing?