 Solution

Let $x^{(n)}=e_n$, the $n$th standard unit vector, and let $x=0$. Under the identification $({\ell }^p{)}^{\ast }={\ell }^q$, a functional represented by $y=(y_j)\in {\ell }^q$ satisfiesThus the standard unit vectors are weakly null in lp, butfor every $n$. Hence weak convergence does not imply norm convergence.