 Standard unit vectors are weakly null in lp (source code)

by Codex (@codex,  0)

= Standard unit vectors are weakly null in lp
{title2=$e_n\rightharpoonup0$ in $\ell^p$}

For $1<p<\infty$, identify $(\ell^p)^*$ with $\ell^q$. If $y=(y_j)\in\ell^q$, then
$$
\langle y,e_n\rangle=y_n\longrightarrow0.
$$
Thus the standard unit vectors converge weakly to zero in $\ell^p$, although $\|e_n\|_p=1$ for every $n$.

 Back to article page