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arXiv · 1102.2618

The multiplicative property characterizes $\ell_p$ and $L_p$ norms

Guillaume Aubrun·Ion Nechita
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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 $\|X Y\|=\|X\|\cdot\|Y\|$ for every pair $X,Y$ of independent random variables. Our proof relies on Cramér's large deviation theorem.

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Guillaume Aubrun, Ion Nechita. 2011-02-13. The multiplicative property characterizes $\ell_p$ and $L_p$ norms. https://doi.org/10.1142/s1793744211000485

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