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A fact I have not been able to locate regarding nets:

If I have a net $\{a_{\alpha}\}$, and fix a nonzero vector $y$ in a topological vector space $X$, and specify that $\{a_{\alpha}y\}$ converges to $x \in X$, then does the original net $\{a_{\alpha}\}$ necessarily have a convergent subnet if it is eventually bounded?

EDIT: It appears the eventual boundedness of the net $\{a_{\alpha}\}$ follows from the given conditions, so perhaps the question could be better phrased as "Does every eventually bounded net in $\Bbb C^{n}$ have a convergent subnet?"

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  • $\begingroup$ Well, a net certainly need not be a sequence. For example, consider $a_\alpha=(1-\alpha)\vec y$ for some $\vec y\ne\vec 0,$ where $\alpha\in[0,1].$ A net is simply a function on a directed set. A net in $\Bbb C^n$ is an $\Bbb C^n$-valued function on a directed set. $\endgroup$ – Cameron Buie Sep 9 '15 at 23:33
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Yes. If $\{a_\alpha\}\subseteq \mathbb{C}^n$ is eventually bounded, we may assume that it is bounded. Then the existence of a convergent subnet follows by compactness of $\mathbb{C}^n$.

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    $\begingroup$ How is $\Bbb C^n$ compact? $\endgroup$ – user269374 Sep 9 '15 at 23:57
  • $\begingroup$ I assume you mean compactness of a closed bounded $\Bbb C^n$-ball? $\endgroup$ – Cameron Buie Sep 9 '15 at 23:59
  • $\begingroup$ Yes, my mistake. $\endgroup$ – Andy Sep 10 '15 at 21:48

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