Random Banach spaces. The limitations of the method

Abstract

We study the properties of "generic", in the sense of the Haar measure on the corresponding Grassmann manifold, subspaces of l^N_infinity of given dimension. We prove that every "well bounded" operator on such a subspace, say E, is a "small" perturbation of a multiple of identity, where "smallness" is defined intrinsically in terms of the geometry of E. In the opposite direction, we prove that such "generic subspaces of l^N_infinity" do admit "nontrivial well bounded" projections, which shows the "near optimality" of the first mentioned result, and proves the so called "Pisier's dichotomy conjecture" in the "generic" case.