Talk:Dvoretzky's theorem

Different formulations
I've stumbled upon a formulation of Dvoretzky theorem which uses Banach-Mazur distance.

Theorem 6.2.1 in the book Kadets, Kadets: Series in Banach spaces
 * Let k be an arbitrary natural number and let $$\varepsilon>0$$. Then there exists a number $$n(k,\varepsilon)$$ such that, for any normed space X with $$dim X>n(k,\varepsilon)$$ there is a k-dimensional subspace Y of X such that $$d\left(Y,l_2^{(k)}\right)<1+\varepsilon$$.

However as am far from being expert in this area, I do not know whether this formulation is equivalent to the one given in the article. And I definitely do not feel competent enough to say whether this is interesting enough to be included in the article. --Kompik (talk) 12:37, 24 February 2011 (UTC)


 * ✅ This formulation is equivalent; I have added it to the article. AxelBoldt (talk) 00:10, 9 January 2017 (UTC)

Question
In the section "Further Development", it says
 * More precisely, let Sn &minus; 1 denote the unit sphere with respect to some Euclidean structure Q [...] For any Q, there exists such a subspace E

Is Q here a quadratic form on X or on E? I assume it lives on X, so Sn &minus; 1 should be written as SN &minus; 1. If this is correct, are we considering the Euclidean norm on E that is induced by Q|E? AxelBoldt (talk) 03:08, 1 January 2017 (UTC)


 * ✅ I have figured it out and edited the article accordingly. AxelBoldt (talk) 00:10, 9 January 2017 (UTC)