  In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n
\subset L_2$ that is bounded in $L_{\infty}$, 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 $L_1$ 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.
