The multiplicative property characterizes ℓₚ and Lₚ norms

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We show that ℓₚ norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the Lₚ norms are the unique rearrangement-invariant norms on a probability space such that X Y=X·Y for every pair X,Y of independent random variables. Our proof relies on Cramér's large deviation theorem.