TITLE: Extension of the limit operator on $l^\infty$
QUESTION [5 upvotes]: Let $l^\infty = \{x\in \mathbb{R}^\mathbb{N}\colon \sup_{n\in \mathbb{N}}|x_n|<\infty\}$
and the subspace $C \subseteq l^\infty$ given by the convergent sequences. 
We consider the linear operator $L$ in $C$ given by $$C \ni x \mapsto L(x) =\lim_nx_n$$
Is easy to see that $L$ is continuous with supremum norm hence by Hahn Banach extension theorem there exists $L^* \in (l^\infty)^*$ which is an extension of $L$.
There exists a explicit form for such extension?
Thanks!

REPLY [3 votes]: There is not. The closest you can come to explicitness, so far as I know, is to let $p$ be a free ultrafilter on $\Bbb N$ and extend $L$ to the $p$-limit: $p$-$\lim_nx_n=a$ iff for all $\epsilon>0$ $\{n\in\Bbb N:|x_n-a|<\epsilon\}\in p$.