In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman.[1]
Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds:
The quotient space E = Y / Z is of dimension dim E ≥ cN, where c > 0 is a universal constant.
The induced norm || · || on E, defined by
is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that
for
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
[2]
^The original proof appeared in Milman (1984). See also Pisier (1989).
^See references for improved estimates.
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