Let $X$ be a topological vector space. For a countable subset subset $(a_n)_n\subset X$, can one describe the closed convex hull of $(a_n)_n$ as the set $\{\sum_{n=1}^{\infty}c_na_n: 0\leq c_n\leq 1, \ \text{and} \ \sum_{n=1}^{\infty}c_n=1\}$ where the series converges?
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1Possibly related: https://math.stackexchange.com/q/1915689/42969. – Martin R Feb 03 '19 at 07:03
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@MartinR In your counter example, I think the convex set is not (norm) closed. – user124910 Feb 03 '19 at 07:48
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No.
Let $X=\Bbb R$ and $a_n=1/n$ for $n\ge1$. The closed convex hull of the $a_n$ is $[0,1]$, but it's clear that $0\ne\sum c_na_n$ with $c_n\ge0$, $\sum c_n=1$.
David C. Ullrich
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