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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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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 $c_n\ge0$, $\sum c_n=1$.

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