Nemirovski's Inequalities Revisited

What happens if the Xi ’s take values in a (real) Banach space (B, ‖ · ‖)? In such cases, in particular when the square of the norm ‖ · ‖ is not given by an inner product, we are aiming at inequalities of the following type: Let X1, X2, . . . , Xn be independent random vectors with values in (B, ‖ · ‖) with EXi = 0 and E‖Xi‖2 < ∞. With Sn := ∑n i=1 Xi we want to show that E‖Sn‖ ≤ K n ∑ i=1 E‖Xi‖ (4)