Hello,
I am trying to show that every metrizable locally compact topological group admits a complete metric generating the topology of the group
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2$\begingroup$ Could you please give some background or motivation (why do you want to know? why do you think it's true? for which examples do you know it's true? etc) $\endgroup$– Yemon ChoiCommented Aug 15, 2011 at 20:41
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$\begingroup$ suggestions: Choose one side, say left. Show that the left uniformity is complete. Show that a left-invariant metric exists. Relate these two. $\endgroup$– Gerald EdgarCommented Aug 15, 2011 at 21:24
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$\begingroup$ If you make the completion of the metric space (taking the Cauchy sequences) this is coherent by algebraic group operations $\endgroup$– Buschi SergioCommented Aug 15, 2011 at 21:56
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1 Answer
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Every second countable, locally compact group admits a metric which is left-invariant, generates the topology, and is proper (i.e. closed balls are compact). See Theorem 4.5 in http://arxiv.org/pdf/math/0606794
Such a metric is clearly complete.
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$\begingroup$ However, the solution for the homework-type problem proposed was known long before 2006. $\endgroup$ Commented Aug 16, 2011 at 12:06
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2$\begingroup$ @Gerald: Indeed, Yves Cornulier just pointed out to me this paper: Struble, Raimond A. Metrics in locally compact groups. Compositio Mathematica, 28 no. 3 (1974), p. 217-222 numdam.org/numdam-bin/fitem?id=CM_1974__28_3_217_0 $\endgroup$ Commented Aug 16, 2011 at 14:36
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$\begingroup$ A related result from 1951: "Invariant Metrics in Groups" by V.L. Klee Jr., in which it is shown that every Abelian, completely metrizable group admits a complete invariant metric. $\endgroup$– Alex M.Commented Jan 25, 2018 at 15:46
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$\begingroup$ @AlainValette I cannot see the link anymore...to the numdam paper, do you know where it has traveled? Merci en avance :) $\endgroup$– ABIMCommented Jan 22, 2019 at 14:37