Timeline for Is a normed space which is homeomorphic to a Banach space complete?

7 events

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Dec 8, 2016 at 13:06	comment	added	Neslihan		@Gro-Tsen: No, I dont think this argument works since my homeomorphism need not be a group isomorphism, so it need not be a morphism of the uniform spaces, in particular the complete topological vector space $\mathbb R$ is homeomorphic to the noncomplete uniform space $ \left] 0 , 1 \right[$ which is not complete ...
Dec 7, 2016 at 23:13	comment	added	Gro-Tsen		I was about to make the following answer: in a topological group, there is a natural uniform structure (Bourbaki, TG, III, §3), so in particular, in a topological vector space, topology determines a uniform structure; now completeness is a uniform property (viz., all Cauchy ultrafilters are convergent), so in topological groups it must be preserved by homeomorphisms. But now I've completely confused myself as to whether this answer is correct or not!
Dec 7, 2016 at 14:58	vote	accept	Neslihan
Dec 7, 2016 at 14:48	answer	added	Jochen Wengenroth		timeline score: 27
Dec 7, 2016 at 14:46	answer	added	Nate Eldredge		timeline score: 32
Dec 7, 2016 at 14:39	history	edited	YCor		edited tags
Dec 7, 2016 at 13:55	history	asked	Neslihan	CC BY-SA 3.0