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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.

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    @Norbert I may missing something obvious, but why such functional implies the above inequalities?2012-07-04
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    here is a proof http://homepages.math.uic.edu/~furman/math569/Banach-LIM.pdf2012-07-04
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    thanks that's seems intersting!2012-07-04
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    Not at all ;)${}$2012-07-04

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