Can a sequence $\{x_n\}$ in a normed linear space satisfies that $\lim{l(x_n)} = l(x)$ for a set dense in $X'$,but not weakly converge? Suppose that a sequence $\{x_n\}$ of points in a normed linear space satisfies that $\lim{l(x_n)} = l(x)$ for a set $A$ of $l$ dense in  $X'$(the dual of $X$,  the collection of all continuous linear functionals, with the norm $|f|_{\infty} = \sup_{X}|f(x)|$).
Is there an example such that the sequence $x_n$ doesn't weakly converge to $x$, which is, there is $g \in X'$ such that $\lim{g(x_n)} \neq g(x)$? 

Take $X:=\ell^2$ the space of real square summable sequences and $x_n(k):=n\delta_{nk}$, where $\delta_{nk}=0$ if $n\neq k$ and $1$ otherwise. Consider $A$ the space of sequences with finitely many non-zero terms. Then $\langle l;x_n\rangle\to 0$ for all $l\in A$, but $\{x_n\}$ is not bounded.
However, we could conclude weak convergence if we had add boundedness of $\{x_n\}$.
