 
Summary: THE MULTIPLICATIVE PROPERTY CHARACTERIZES p AND Lp
NORMS
GUILLAUME AUBRUN AND ION NECHITA
Abstract. We show that p 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 rearrangementinvariant norms on a probability space
such that XY = X · Y for every pair X, Y of independent random variables. Our proof
relies on Cram´er's large deviation theorem.
1. Introduction
The p and Lp spaces are among the most important examples of Banach spaces and
have been widely investigated (see e.g. [1] for a survey). In this note, we show a new
characterization of the p/Lp norms by a simple algebraic identity: the multiplicative property.
In the case of p norms, this property reads as x y = x · y for every (finite) sequences
x, y. In the case of Lp norms, it becomes XY = X · Y whenever X, Y are independent
random variables.
Inspiration for the present note comes from quantum information theory, where the mul
tiplicative property of the commutative and noncommutative p norms plays an important
role; see [7, 2] and references therein.
1.1. Discrete case: characterization of p norms. Let c00 be the space of finitely sup
ported real sequences. The coordinates of an element x c00 are denoted (xi)iN . If
