Subspaces and orthogonal decompositions generated by bounded orthogonal systems

We investigate properties of subspaces of L ₂ spanned by subsets of a finite orthonormal system bounded in the L _∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L ₁ and the L ₂ norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L ₂ ⁿ, complementary to each other and each of dimension roughly n/2, spanned by ± 1 vectors (i.e. Kashin's splitting) and in logarithmic distance to the Euclidean space. The same method applies for p > 2, and, in connection with the Λ _p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L ₂ and the L _p norms are close (again, up to a logarithmic factor).