In this article we prove that for any orthonormal system (\vphi_j)_{j=1}^n
\subset L_2 that is bounded in L∞, and any 1<k<n, there exists
a subset I of cardinality greater than n−k such that on \spa\{\vphi_i\}_{i
\in I}, the L1 norm and the L2 norm are equivalent up to a factor μ(logμ)5/2, where μ=n/klogk. 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