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Given a compact subset $A$ of a Banach lattice $E$, is the following true?

There exist $u,v\in E$ so that $u\leq a\leq v$ for all $a\in A$.

This is true in case $E=C(X)$, $X$ compact, with the uniform norm. One way of disproving this conjecture is to construct a norm convergent sequence in $E$ which is not order bounded. Probably one would have to consider some $E=L^{p}$, but I've failed to come up with an example. I've also looked in some standard sources, such as Zaanen's "Riesz spaces, vol II" and Schaefer's "Banach lattices and positive operators", but could not find anything.

Any suggestions would be greatly appreciated.

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One way of disproving this conjecture is to construct a norm convergent sequence in E which is not order bounded.

E.g. the sequence $(\frac{1}{n}e_n)$ in $l^1$.

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I feel like I should add that there exists a nice characterization due to Tony Wickstead of ordered Banach spaces whose compact sets are order bounded.

Wickstead A.W., Compact subsets of partially ordered Banach spaces, Mathematische Annalen (1975) (MSN)

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