In this article we prove that for any orthonormal system that is bounded in , and any , there exists a subset of cardinality greater than such that on , the norm and the norm are equivalent up to a factor , where . 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.
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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. 1075–1095DOI 10.4171/RMI/567