Majorizing measures and proportional subsets of bounded orthonormal systems

Abstract

In this article we prove that for any orthonormal system $({\phi }_j{)}_{j=1}^n\subset L_2$ 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 $\mathrm{span}\{{\phi }_i{\}}_{i\in I}$, 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.