 Standard unit vectors are weakly null in lp

For $1<p<\infty$, identify $({\ell }^p{)}^{\ast }$ with ${\ell }^q$. If $y=(y_j)\in {\ell }^q$, thenThus the standard unit vectors converge weakly to zero in ${\ell }^p$, although $\Vert e_n{\Vert }_p=1$ for every $n$.