Weak convergence and weak* convergence in $l^\infty$

Question

I'm looking for a sequence that converges weakly* in $l^\infty$ but not weakly.

In $l^1$ I can take $(e_n)_n$ with $e_n(k)=\delta_{k,n}$, but I don't know if there is a simple example in $l^\infty$. Thank you!

Answer 1

Let $$ x_n = (0,\dots, 0, 1,1, \dots), $$ so the first $n$ elements are zero, all others are $1$. Then $x_n \rightharpoonup^* 0 $ in $l^\infty =(l_1)^*$, but $x_n \not\rightharpoonup 0$: take any Hahn-Banach extension of the limit functional ($f(x) =\lim_{n\to\infty}x_n$ for $x_n\in c$), then $f(x_n) =1 \not\to 0 = f(x)$.