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 $dimX>n(k,\varepsilon )$ 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)

 Done 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 S^n − 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 S^n − 1 should be written as S^N − 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)

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