# Closed Convex Hull

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?

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