# Weak convergence of $\Bigl(\sum\limits_{k=n}^\infty e_k\Bigr)_n$

Let $$e_n:=δ_{kn}$$, for $$k\in\mathbb N$$. Given the sequence $$(a_n):=\sum\limits_{k=n}^∞e_k\subset\ell^\infty$$, i.e.$$((1,1,\cdots),(0,1,1,\cdots),(0,0,1,1,\cdots),\cdots).$$ I want to know if $$(a_n)$$ converges weakly to zero.

Assume $$(a_n)$$ does not converges weakly, then I could use Hahn-Banach to find a linear functional $$\varphi \in (\ell^\infty)^*$$ with $$\lim_{n \to \infty} \varphi((a_n)) \neq 0 \; ,$$ but if $$(a_n)$$ converges weakly to zero, then I have no idea to show this, since I lack an usefull characterization of $$(\ell^\infty)^*$$.

It is possible to use Banach limits. A Banach limit $$\phi$$ is in particular a linear continuous functional on $$\ell^\infty$$ such that if $$x\in \ell^\infty$$ converges to some $$\ell$$, then $$\phi(x)=\ell$$. Here $$\phi(a_n)=1$$ hence there cannot be weak convergence to zero.
Actually, there is no weak convergence at all since using the evaluation maps $$e_n\colon (x_k)_{k\geqslant 1}\in\ell^\infty\mapsto x_n$$, the only potential weak limit is the null sequence.
• Regarding your last sentence: By convergence to zero I mean convergence to the zero sequence (i.e. the zero of the vector space $\ell^\infty$). May 12, 2020 at 14:36