Quotient of subspace theorem

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 ≥ c N, where c > 0 is a universal constant.
• The induced norm || · || on E, defined by

${\displaystyle \|e\|=\min _{y\in e}\|y\|,\quad e\in E,}$

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

${\displaystyle {\frac {\sqrt {Q(e)}}{K}}\leq \|e\|\leq K{\sqrt {Q(e)}}}$ for ${\displaystyle e\in E,}$

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

${\displaystyle c(K)\approx 1-{\text{const}}/\log \log K.}$^[2]

Notes

1. The original proof appeared in Milman (1984). See also Pisier (1989).
2. See references for improved estimates.

References

• Milman, V.D. (1984), "Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space", Israel seminar on geometrical aspects of functional analysis, Tel Aviv: Tel Aviv Univ., X
• Gordon, Y. (1988), "On Milman's inequality and random subspaces which escape through a mesh in Rⁿ", Geometric aspects of functional analysis, Lecture Notes in Math., Berlin: Springer, 1317: 84–106, doi:10.1007/BFb0081737, ISBN 978-3-540-19353-1
• Pisier, G. (1989), The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, 94, Cambridge: Cambridge University Press