# Existence of a $l_ {\infty} ^*$ element [duplicate]

Exercise: Prove there exist a bounded linear functional $L :l_{ \infty} \rightarrow \mathbb{R}$ such that for every $(x_n)=x \in l _ { \infty }$ $$\lim \mathrm{Inf} (x_n) \leq L(x) \leq \lim \mathrm{Sup} (x_n).$$ My current progress is that should be $L \in l _{ \infty}^*\setminus l_1$. Also I know since $l_1$ is not bidual so there are such functionals. Any help is appreciated.

## marked as duplicate by Norbert, Asaf Karagila♦, t.b., Davide Giraudo, J. M. is a poor mathematicianJul 10 '12 at 4:06

• Not at all ;)${}$ – Norbert Jul 4 '12 at 18:50