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?


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.

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  • $\begingroup$ Good answer though it mostly shows that I did not properly generalize my actual problem... I decided to state the exact case that I am interested in in a separate question (mathoverflow.net/q/332478/101775). $\endgroup$ – Dominique Unruh May 25 '19 at 23:14
  • $\begingroup$ A finite-dimensional example is the cone consisting of $0$ and the points $(x,y)$ with $x > 0$ and $y > 0$ in $\mathbb{R}^2$. $\endgroup$ – Robert Furber May 26 '19 at 20:30

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