Quotient of subspace theorem

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:

• The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0 is a universal constant.
• The induced norm || · || on E, defined by

$\Vert e\Vert =\underset{y\in e}{\min }\Vert y\Vert ,e\in E,$

is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that

$\frac{\sqrt{Q(e)}}{K}\le \Vert e\Vert \le K\sqrt{Q(e)}$ for $e\in E,$

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

$c(K)\approx 1-\text{const}/\log \log K.$

References

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