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?