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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 rearrangement-invariant 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
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