Abstract
In this article we prove that for any orthonormal system
(\varphi_j)_{j=1}^n \subset L_2
that is bounded in
L_{\infty}
, and any
1
, there exists a subset
I
of cardinality greater than
n-k
such that on
\mathrm{span}\{\varphi_i\}_{i \in I}
, the
L_1
norm and the
L_2
norm are equivalent up to a factor
\mu (\log \mu)^{5/2}
, where
\mu = \sqrt{n/k} \sqrt{\log k}
. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.