# Closed convex cone - equivalence of definition via closure and via infinite sums

I have a set $$P$$ of points in a Banach space. Consider the following two cones:

• The closure of the set of all (finite) nonnegative linear combinations of $$P$$. (I.e., the topological closure of $$\{\sum_{i=1}^n a_ip_i: a_i\geq0, p_i\in P\}$$.)
• The set of all infinite nonnegative linear combinations of $$P$$. (I.e., $$\{\sum_i a_ip_i: a_i\geq0, p_i\in P\}$$ where $$i$$ can range over infinite sets, and we only consider sums that converge absolutely.)

Are those sets equal?

Example showing that the second set can be strictly smaller: Denote by $$\{e_n\}$$ the unit vector basis in $$\ell_1$$ and consider the following set $$P:=\{e_1+\frac1ne_n\}_{n=2}^\infty$$ in $$\ell_1$$. It is clear that $$e_1$$ is in the first set, but not in the second.
• A finite-dimensional example is the cone consisting of $0$ and the points $(x,y)$ with $x > 0$ and $y > 0$ in $\mathbb{R}^2$. – Robert Furber May 26 '19 at 20:30