Talk:Euclidean Space is Banach Space

I'm hopelessly confused -- is or is not a Banach space a normed complete metric space or just a complete metric space? It has to be one or the other. --prime mover (talk) 14:33, 12 May 2020 (EDT)


 * Reverted back to normed vector space. I myself got confused because I saw cases e.g. Euclidean Space is Banach Space, where Complete Metric Space was concluded to be Banach Space. So I assumed that in some sense Banach space could be generalized. But since it is not the case (Banach space is subset of Complete Metric Space), I will relink articles I modified, and we have to check where else complete metric space jumps to conclusion. --Julius (talk) 16:08, 12 May 2020 (EDT)


 * Ok, I see that there is also Euclidean Space is Normed Space. But then, strictly speaking, we need one more short step to show that completeness from metric space gets transferred to normed vector space. In general, I find these similarities between metric and normed spaces very slippery.--Julius (talk) 16:18, 12 May 2020 (EDT)


 * From having looked at it, this area seems already to be pretty complete: you have a complete metric space, which is what it is, and then you have a complete metric space which has a norm on it, which is then a complete normed metric space, but we don't call it that, we call it a Banach space. So we have Euclidean space is complete metric space as one result, and then when you establish that there is in fact a norm on a Euclidean space, you then prove it is a Banach space by first proving the norm is what it is, then you invoke the proof that a Euclidean space is a complete metric space. Job done, move on. --prime mover (talk) 17:05, 12 May 2020 (EDT)