Quotient of subspace theorem

From Wikipedia, the free encyclopedia

The quotient of subspace theorem is an important property of finite dimensional normed spaces, discovered by Vitali Milman.

Let (X, \| \cdot \|) be an N-dimensional normed space. There exist subspaces Z \subset Y \subset X such that the following holds:

  • The quotient space E = Y / Z is of dimension \text{dim} E \geq c N, where c > 0 is a universal constant.
  • The induced norm \| \cdot \| on E, defined by \| e \| = \min_{y \in e} \| y \| for e \in E, is isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that
\frac{\sqrt{Q(e)}}{K} \leq \| e \| \leq K \sqrt{Q(e)} for e \in E,
with K > 1 a universal constant.

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 ;

see references for improved estimates.

[edit] References

  • V.D.Milman, Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space, Israel seminar on geometrical aspects of functional analysis (1983/84), X, 8 pp., Tel Aviv Univ., Tel Aviv, 1984.
  • Y. Gordon, On Milman's inequality and random subspaces which escape through a mesh in $R\sp n$, Geometric aspects of functional analysis (1986/87), 84--106, Lecture Notes in Math., 1317, Springer, Berlin, 1988.
  • G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989. xvi+250 pp.