Weakly null sequence

A sequence $(x_n)$ is weakly null when $f(x_n)\to 0$ for every member $f$ of the continuous dual space. It may remain bounded away from zero in norm.

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 $n$.

Let $1<p<\infty$. Every norm-bounded sequence in ${\ell }^p$ whose terms have pairwise disjoint supports converges weakly to zero. Indeed, the average of $N$ distinct terms has norm $O(N^{1/p-1})$, so Mazur theorem gives weak convergence through its convex-combination criterion.