Existence of a $ l_ {\infty} ^*$ element

Asked Jul 04 '12 at 17:36

Active Jul 04 '12 at 17:51

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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.

edited Apr 13 '17 at 12:19

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asked Jul 04 '12 at 17:36

clark

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@Norbert I may missing something obvious, but why such functional implies the above inequalities? – clark Jul 04 '12 at 18:41
here is a proof http://homepages.math.uic.edu/~furman/math569/Banach-LIM.pdf – Norbert Jul 04 '12 at 18:44
thanks that's seems intersting! – clark Jul 04 '12 at 18:50
1

Not at all ;)${}$ – Norbert Jul 04 '12 at 18:50

Existence of a $ l_ {\infty} ^*$ element

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