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Is there a characterization of nets with no convergent subnets in a Banach space?

Does someone know some survey or book's chapter that deals with this kind of stuff?

Martin Sleziak
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André Porto
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  • https://math.stackexchange.com/questions/1209341/example-of-converging-subnet-when-there-is-no-converging-subsequence this might help – Arpan1729 May 10 '17 at 03:27
  • Strong topology in a Banach space is metric, so convergence is fully characterized by sequences. You are probably referring to weak or $*$-weak topologies, or maybe some operator topologies, which are non-metric, but there is not enough context in the post to answer specifically. – Conifold May 10 '17 at 04:32
  • Actually, I'm aware of weak topologies in Banach spaces and of course the E1 property of metric spaces. Just consider the problem. Let us work with sequences first then. When the Banach space is finite dimensional, its ball is compact, so a sequence which has no convergent subsequence must have no limited subsequence, that is, $$liminf |x_n|=\infty.$$ Of course, the converse is true, so this formula characterizes sequences with no convergent subsequences in finite dimensional Banach spaces. Applying the analogue of this formula to nets, we get this characterization for nets. – André Porto May 10 '17 at 08:48
  • But is there a characterization for infinite-dimensional Banach spaces? We could consider an analogue question for complete metric spaces in general, "Is there a characterization of nets such that none of its subsequences (subnets directed by $\omega$) converges?" – André Porto May 10 '17 at 08:59
  • Somewhat more detailed question is now on MO: Characterization of nets with no convergent subnets in Banach spaces – Martin Sleziak May 12 '17 at 04:55
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    Some basic info about cross-posting: https://math.meta.stackexchange.com/tags/cross-posting/info Especially the suggestions in this answer seem very reasonable to me. – Martin Sleziak May 12 '17 at 04:57

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