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 $1 < k < n$, 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.