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Referring to the paper at http://arxiv.org/abs/0910.0608. In the proof of lemma three, can any one explain why the unit sphere Sv that be founded in the end of the proof is euclidean? and also if it so, why the normed space V must be euclidean (please see the last step is saying "Thus the unit sphere of V is a euclidean sphere, and V must be eulidean)? I thanks in advance any one can explain and simplify this to me as soon as possible.

Note: I have asked this question a few hours ago but may be in a wrong way so the people here may wouldn't understand the question, so if they just re read the question again and answer it me.

Thanks for every one who will try to help me.

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possible duplicate of Euclidean Sphere –  amWhy Mar 4 '13 at 4:40
It is best to edit an earlier post, for example, with the link you've included here, than it is to re-post the same question, with edit. –  amWhy Mar 4 '13 at 4:41
Dear user50382: as amWhy (and you) notice, this is essentially the same question you posted earlier today. However, this version is a bit more detailed. As reposting the same question is discouraged, I suggest you consider deleting the old question. –  gnometorule Mar 4 '13 at 4:41
I am sorry if I didn't delete the previous one, but just I tried to re write the question properly as a new question because I didn't know that I should delete the old one. so please I still waiting for the answer and I am in desperate need to it. –  LoveMath Mar 4 '13 at 5:38
Hint: Reconsider your practice of pleading on urgency. It gives the problem the air of homework due in a couple of hours. Homework problems are welcome here, but should be honestly tagged as such. At best, such pleading will have zero effect. At worst, it may be a turn off for people who might be able to answer. –  Jyrki Lahtonen Mar 4 '13 at 6:30

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