In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman.
Let ("X", ||·||) be an "N"-dimensional normed space. There exist subspaces "Z" ⊂ "Y" ⊂ "X" such that the following holds:
 formula_1
is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") "Q" on "E", such that
 formula_2 for formula_3
with "K"  ; 1 a universal constant. 
The statement is relative easy to prove by induction on the dimension of "Z" (even for "Y=Z", "X"="0", "c=1") with a "K" that depends only on "N"; the point of the theorem is that "K" is independent of "N".
In fact, the constant "c" can be made arbitrarily close to 1, at the expense of the
constant "K" becoming large. The original proof allowed 
formula_4
Notes.
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