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Nonnegative linear functionals over $l^\infty$
An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$?

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 mathematician Jul 10 '12 at 4:06

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  • $\begingroup$ @Norbert I may missing something obvious, but why such functional implies the above inequalities? $\endgroup$ – clark Jul 4 '12 at 18:41
  • $\begingroup$ here is a proof homepages.math.uic.edu/~furman/math569/Banach-LIM.pdf $\endgroup$ – Norbert Jul 4 '12 at 18:44
  • $\begingroup$ thanks that's seems intersting! $\endgroup$ – clark Jul 4 '12 at 18:50
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    $\begingroup$ Not at all ;)${}$ $\endgroup$ – Norbert Jul 4 '12 at 18:50