Let $X$ be a vector space containing a convex cone $C$ which induces a (pre) order on $X$ by $x \geq y \Leftrightarrow x-y\in C$. Given a sequence of linear functionals $x^*_n:X\rightarrow\mathbb{R}$ converging to 0 pointwise (i.e. in the weak* topology), under what conditions can one claim that $$x^*_n\vee 0(x):=\sup\{x^*_n(y);0\leq y\leq x\} \rightarrow0$$ as $n\rightarrow \infty $ for any $x \geq 0$?

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    My answer to this recent question gives an easy counterexample when $X = L^1[0,1]$. On the other hand, it's true for $X = l^1$. "Under what conditions" is too vague to guide us to an answer that would help you with the specific issue that led you to ask this. – Nik Weaver Nov 26 at 20:04

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