The multiplicative property characterizes ${\ell }_p$ and $L_p$ norms

Abstract

We show that ${\ell }_p$ norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the $L_p$ norms are the unique rearrangement-invariant norms on a probability space such that $\Vert XY\Vert =\Vert X\Vert \cdot \Vert Y\Vert$ for every pair $X,Y$ of independent random variables. Our proof relies on Cramér's large deviation theorem.

8 pages, 1 figure

References in corpus (2)

• Catalytic majorization and ${\ell }_p$ norms
• Stochastic domination for iterated convolutions and catalytic majorization

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