# Majorizing measures and proportional subsets of bounded orthonormal systems

### Olivier Guédon

Université Paris-Est, Marne-la-Vallée, France### Shahar Mendelson

Technion - Israel Institute of Technology, Haifa, Israel### Alain Pajor

Université Paris-Est, Marne-la-Vallée France### Nicole Tomczak-Jaegermann

University of Alberta, Edmonton, Canada

## 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 $1 < k < n$, 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.

## Cite this article

Olivier Guédon, Shahar Mendelson, Alain Pajor, Nicole Tomczak-Jaegermann, Majorizing measures and proportional subsets of bounded orthonormal systems. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 1075–1095

DOI 10.4171/RMI/567