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?



Closed convex cone - equivalence of definition via closure and via infinite sums
The answer to your question is "No" (but the first set, obviously, always contains the second). 


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.


