Abstract
We investigate properties of subspaces of L2 spanned by subsets of a flnite orthonormal system bounded in the L1 norm. We flrst prove that there exists an arbitrarily large subset of this orthonormal system on which the L1 and the L2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L n 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 ⁄p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L2 and the Lp norms are close (again, up to a logarithmic factor).
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