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 ‖XY‖ = ‖X‖ ⋅ ‖Y‖ for every pair X, Y of independent random variables. Our proof combines the tensor power trick and Cramér's large deviation theorem.
@article{CML_2011__3_4_637_0,
author = {Aubrun, Guillaume and Nechita, Ion},
title = {The multiplicative property characterizes $\ell _p$ and $L_p$ norms},
journal = {Confluentes Mathematici},
pages = {637--647},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {4},
year = {2011},
doi = {10.1142/S1793744211000485},
language = {en},
url = {http://archive.numdam.org/articles/10.1142/S1793744211000485/}
}
TY - JOUR AU - Aubrun, Guillaume AU - Nechita, Ion TI - The multiplicative property characterizes $\ell _p$ and $L_p$ norms JO - Confluentes Mathematici PY - 2011 SP - 637 EP - 647 VL - 3 IS - 4 PB - World Scientific Publishing Co Pte Ltd UR - http://archive.numdam.org/articles/10.1142/S1793744211000485/ DO - 10.1142/S1793744211000485 LA - en ID - CML_2011__3_4_637_0 ER -
%0 Journal Article %A Aubrun, Guillaume %A Nechita, Ion %T The multiplicative property characterizes $\ell _p$ and $L_p$ norms %J Confluentes Mathematici %D 2011 %P 637-647 %V 3 %N 4 %I World Scientific Publishing Co Pte Ltd %U http://archive.numdam.org/articles/10.1142/S1793744211000485/ %R 10.1142/S1793744211000485 %G en %F CML_2011__3_4_637_0
Aubrun, Guillaume; Nechita, Ion. The multiplicative property characterizes $\ell _p$ and $L_p$ norms. Confluentes Mathematici, Tome 3 (2011) no. 4, pp. 637-647. doi : 10.1142/S1793744211000485. http://archive.numdam.org/articles/10.1142/S1793744211000485/
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