The Rademacher Projection

Abstract

In this chapter we return to the subject of estimating the norm of Rad _n — the natural projection onto the subspace {Σ ⁿ_i=1 r _i (t)x _i ; x _i ∈X} of L ₂ (X). The main theorem here (14.5), due to Pisier, states that sup_n<∞ ‖Rad _n ‖ < ∞ if and only if ℓ₁ is not finitely representable in X. Towards the end of the chapter we give also an estimate for ‖Rad _n ‖ good for any finite dimensional normed space. In this chapter it will be more convenient for us to work with complex scalars so we assume all spaces are over the complex field ℂ. (The results are easily seen to hold also in the real case)