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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?

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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
    Commented May 25, 2019 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
    Commented May 26, 2019 at 20:30

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