Quotient of subspace theorem (original) (raw)
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 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.