# Continuity of the positive part of a linear functional

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 $$x \geq 0$$?

• 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