THE MULTIPLICATIVE PROPERTY CHARACTERIZES ℓ_{p} AND L_{p} NORMS

GUILLAUME AUBRUN AND ION NECHITA

Abstract. We show that ℓ_{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 ∥XY ∥ = ∥X∥·∥Y ∥ for every pair X, Y of independent random variables. Our proof relies on Cramér’s large deviation theorem.

1. Introduction

The ℓ_{p} and L_{p} spaces are among the most important examples of Banach spaces and have been widely investigated (see e.g. [2] for a survey). In this note, we show a new characterization of the ℓ_{p}/L_{p} norms by a simple algebraic identity: the multiplicative property. In the case of ℓ_{p} norms, this property reads as ∥x ⊗ y∥ = ∥x∥ · ∥y∥ for every (finite) sequences x, y. In the case of L_{p} 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 multiplicative property of the commutative and noncommutative ℓ_{p} norms plays an important role; see [8, 3] and references therein. Also, Theorem 1.1 below can be seen as a partial answer to the open problem below Proposition (5.2.38) in [1].

1.1. Discrete case: characterization of ℓ_{p} norms. Let c_{00} be the space of finitely supported real sequences. The coordinates of an element x ∈ c_{00} are denoted (x_{i})_{i∈N^{∗}}. If x, y ∈ c_{00}, we define x ⊗ y to be double-indexed sequence (x_{i}y_{j})_{(i,j)∈N^{∗}×N^{∗}}. Throughout the paper, we consider x ⊗ y as an element of c_{00} via some fixed bijective map between N^{∗} and N^{∗} × N^{∗}.

We consider a norm ∥ · ∥ on c_{00} satisfying the following conditions

(1) (permutation-invariance) If x, y ∈ c_{00} are equal up to permutation of their coordinates, then ∥x∥ = ∥y∥.

(2) (multiplicativity) If x, y ∈ c_{00}, then ∥x ⊗ y∥ = ∥x∥ · ∥y∥.

Because of the invariance under permutation, the specific choice of a bijection between N^{∗} and N^{∗} × N^{∗} is irrelevant. Examples of a norm satisfying both conditions are given by ℓ_{p} norms, defined by

∥x∥_{p} = ( ∑_{i∈N^{∗}} |x_{i}|^{p})^{1/p} if 1 ⩽ p < +∞ ; ∥x∥_{∞} = sup_{i∈N^{∗}} |x_{i}|.

The next theorem shows that there are no other examples.

Theorem 1.1. If a norm ∥ · ∥ on c_{00} is permutation-invariant and multiplicative, then it coincides with ∥ · ∥_{p} for some p ∈ [1, +∞].

The proof of Theorem 1.1 is simple and goes as follows. First, the value of p is retrieved by looking at ∥(1, 1)∥. Then, for every x ∈ c_{00}, the quantity ∥x∥ is shown to equal ∥x∥_{p} by examining the statistical distribution of large coordinates of the n-th tensor power x^{⊗n} (n large) through Cramér’s large deviations theorem. We defer the proof to section 2.

1.2. Continuous case: characterization of L_{p} norms. We now formulate a version of Theorem 1.1 in a continuous setting, in order to characterize L_{p} norms. Let (Ω, F, P) be a rich probability space, which means that it is possible to define on it one continuous random variable. This implies that we can define on Ω an arbitrary number of independent random variables with arbitrary distributions; one can think of Ω as the interval [0, 1] equipped with the Lebesgue measure. A random variable is said to be simple if it takes only finitely many values. For a random variable X : Ω → R, the L_{p} normd are defined as

∥X∥_{L_{p}} = \begin{cases}(E|X|^{p})^{1/p} & if 1 ⩽ p < +∞, \\ inf\{M s.t. P(|X| ⩽ M) = 1\} & if p = ∞. \end{cases}

The L_{p} norms are rearrangement-invariant (i.e. the norm of a random variable depends only on its distribution) and satisfy the property ∥XY ∥ = ∥X∥·∥Y ∥ whenever X, Y are independent random variables. Note that product of independant random variables correspond to the tensor product in c_{00}. These properties characterize the L_{p} norms:

Theorem 1.2. Let (Ω, F, P) be a rich probability space, and let E be the space of simple random variables. Let ∥ · ∥ be a norm on E with the following properties :

(1) If two random variables X, Y ∈ E have the same distribution, then ∥X∥ = ∥Y ∥,

(2) If two random variables X, Y ∈ E are independent, then ∥XY ∥ = ∥X∥ · ∥Y ∥. Then there exists p ∈ [1, +∞] such that ∥X∥ = ∥X∥_{L_{p}} for every X ∈ E.

We prove Theorem 1.2 in Section 3. We will derive Theorem 1.2 as a consequence of Theorem 1.1. Alternatively one could prove it by mimicking the proof of Theorem 1.1.

2. The case of ℓ_{p} norms: proof of Theorem 1.1

Let ∥.∥ be a norm on c_{00} which is permutation-invariant and multiplicative.

STEP 1. We first show that the norm of an element of c_{00} depends only on the absolute values of its coordinates.

Lemma 2.1. A norm on c_{00} which is permutation-invariant and multiplicative is also unconditional: if x, y ∈ c_{00} have coordinates with equal absolute values (|x_{i}| = |y_{i}| for every i), then ∥x∥ = ∥y∥. As a consequence, if a, b ∈ c_{00} and 0 ⩽ a ⩽ b (coordinatewise), then ∥a∥ ⩽ ∥b∥.

Proof. If x, y have coordinates with equal absolute values, then the vectors x ⊗ (1, −1) and y ⊗ (1, −1) are equal up to permutation of their coordinates. Permutation-invariance and multiplicativity imply that ∥x∥ = ∥y∥. For the second part of the lemma, note that 0 ⩽ a ⩽ b implies that a belongs to the convex hull of the vectors {(ε_{i}b_{i}); ε_{i} = ±1} and use the triangle inequality to conclude. □

Remark 2.2. In the literature, unconditional and permutation-invariant norms are sometimes called symmetric norms.

STEP 2. We now focus on sequences whose nonzero coefficients are equal to 1. We write 1^{n} for the sequence formed with n 1’s followed by infinitely many zeros and we put u_{n} = ∥1^{n}∥. By Lemma 2.1, the sequence (u_{n})_{n} is non-decreasing. Moreover, the multiplicativity property of the norm implies that the sequence (u_{n})_{n} itself is multiplicative: u_{kn} = u_{k}u_{n}. It is folklore that a nonzero non-decreasing sequence (u_{n})_{n} such that u_{kn} = u_{k}u_{n} must equal (n^{α})_{n} for some α ⩾ 0 (see [7] for a proof). We set p = 1/α (p = +∞ if α = 0). By the triangle inequality, u_{n+k} ⩽ u_{n} + u_{k}, which implies that p ⩾ 1. At this point we have proved that

∥1^{n}∥ = n^{1/p}.

To prove Theorem 1.1, we need to show that ∥x∥ = ∥x∥_{p} for every x ∈ c_{00}. The case p = +∞ is easily handled, so we may assume that 1 ⩽ p < +∞. By Lemma 2.1, without loss of generality, we may also assume that the coordinates of x are non-negative and in nonincreasing order. Let k be the number of nonzero coordinates of x ; then x_{i} = 0 for i > k. We will separately show the inequalities ∥x∥ ⩾ ∥x∥_{p} and ∥x∥ ⩽ ∥x∥_{p}. In both cases, we compare x^{⊗n} with simpler vectors and apply Cramér’s theorem (which we now review) to estimate the number of “large” coordinates of x^{⊗n} when n goes to infinity.

Cramér’s theorem. Fix x ∈ c_{00} with non-negative non-increasing coordinates, and let k be the number of nonzero coordinates of x. For a > 0, let N(x, a) be the number of coordinates of x which are larger than or equal to a. To estimate this number, we introduce the convex function Λ_{x} : R → R

Λ_{x}(λ) = ln (∑_{i=1}^{k} x_{i}^{λ})

and its convex conjugate Λ_{x}^{∗} : R → R ∪ {+∞}

Λ_{x}^{∗}(t) = sup_{λ∈R} λt − Λ_{x}(λ).

The Fenchel–Moreau theorem (see e.g. [5]) implies that convex conjugation is an involution: we have, for any λ ∈ R,

Λ_{x}(λ) = sup_{t∈R} λt − Λ_{x}^{∗}(t).

Proposition 2.3 (Cramér’s large deviation theorem). Let x ∈ c_{00} such that x_{i} > 0 for 1 ⩽ i ⩽ k and x_{i} = 0 for i > k. Let t be a real number such that exp(t) ⩽ ∥x∥_{∞}. Then,

lim_{n→∞} \frac{1}{n} ln N(x^{⊗n}, exp(tn)) = \begin{cases}ln k & if exp(t) ⩽ (∏_{i=1}^{k} x_{i})^{1/k} \\ −Λ_{x}^{∗}(t) & otherwise \end{cases} \} ⩾ −Λ_{x}^{∗}(t).

Proof. To see how Proposition 2.3 follows from the standard formulation of Cramér’s theorem, let (X_{n}) be independent random variables with common distribution given by

\frac{1}{k} ∑_{i=1}^{k} δ_{ln x_{i}}.

Then P(\frac{1}{n}(X_{1} + ⋯ + X_{n}) ⩾ t) = \frac{1}{k^{n}} N(x^{⊗n}, exp(tn)). The usual Cramér theorem (see any probability textbook, or [6] for a short proof) asserts that

lim_{n→∞} \frac{1}{n} ln P (\frac{1}{n}(X_{1} + ⋯ + X_{n}) ⩾ t) = \begin{cases}0 & if t ⩽ EX_{1} \\ − sup_{λ∈R} (λt − ln Ee^{λX_{1}}) & otherwise. \end{cases}

This is equivalent to the equality in Proposition 2.3. The last inequality follows easily since Λ_{x}^{∗}(t) ⩾ − ln k for every real t. □

We now complete the proof of the main theorem by comparing x^{⊗n} with simpler vectors, as shown in Figure 1.

Figure 1. Bounding the vector x^{⊗n} by vectors with simpler profiles. The coordinates of the tensor power x^{⊗n} are represented by dark circles, the vector used in for the lower bound has only one non-zero value exp(tn) and the upperbounding vector has values exp(t_{d}n) ⩾ ⋯ ⩾ exp(t_{1}n) ⩾ 0.

STEP 3: the lower bound ∥x∥ ⩾ ∥x∥_{p}. For t ∈ R, we have the lower bound

∥x∥ = ∥x^{⊗n}∥^{1/n} ⩾ ∥ exp(tn)1^{N(x^{⊗n},exp(tn))}∥^{1/n} = exp(t)N(x^{⊗n}, exp(tn))^{1/np}.

Proposition 2.3 asserts that

lim_{n→∞} N(x^{⊗n}, exp(tn))^{1/n} ⩾ exp(−Λ_{x}^{∗}(t)).

We have therefore

∥x∥ ⩾ exp(t − Λ_{x}^{∗}(t)/p) = exp(pt − Λ_{x}^{∗}(t))^{1/p}

for any t ∈ R. Taking the supremum over t and using the Fenchel–Moreau theorem shows that

∥x∥ ⩾ exp(Λ_{x}(p))^{1/p} = ∥x∥_{p}.

STEP 4: The upper bound ∥x∥ ⩽ ∥x∥_{p}. Fix ε > 0 and choose t_{0} < ⋯ < t_{d} such that

exp(t_{0}) = min_{1⩽i⩽k} x_{k}, exp(t_{1}) = (∏_{i=1}^{k} x_{i})^{1/k} , exp(t_{d}) = ∥x∥_{∞} and sup_{2⩽i⩽d} |t_{i} − t_{i−1}| < ε.

For n ∈ N^{∗}, we define a vector y_{n} ∈ c_{00} as follows: the coordinates of y_{n} belong to the set

{0, exp(nt_{1}), exp(nt_{2}), …, exp(nt_{d})}

and are minimal possible such that the inequality x^{⊗n} ⩽ y_{n} holds coordinatewise. Lemma 2.1 implies that ∥x^{⊗n}∥ ⩽ ∥y_{n}∥. On the other hand, for 1 ⩽ i ⩽ d, the number of coordinates of y_{n} which are equal to exp(nt_{i}) is less than N(x^{⊗n}, exp(nt_{i−1})). The triangle inequality implies that

∥y_{n}∥ ⩽ ∑_{i=1}^{d} ∥exp(t_{i}n)1^{N(x^{⊗n},exp(t_{i−1}n))}∥

⩽ ∑_{i=1}^{d} exp(t_{i}n)N(x^{⊗n}, exp(t_{i−1}))^{1/p}
⩽ d max_{1⩽i⩽d} {exp(t_{i}n)N(x^{⊗n}, exp(t_{i−1}n))^{1/p}} .

This gives an upper bound for ∥x∥

(1)  ∥x∥ = ∥x^{⊗n}∥^{1/n} ⩽ ∥y_{n}∥^{1/n} ⩽ d^{1/n} max_{1⩽i⩽d} {exp(t_{i})N(x^{⊗n}, exp(t_{i−1}n))^{1/np}} .

For 2 ⩽ i ⩽ d, Proposition 2.3 implies that

lim_{n→∞} exp(t_{i})N(x^{⊗n}, exp(t_{i−1}n))^{1/np} = exp(t_{i}) exp(−Λ_{x}^{∗}(t_{i−1}))^{1/p}
⩽ exp(t_{i}) exp(−(pt_{i−1} − Λ_{x}(p)))^{1/p}
⩽ exp(ε)∥x∥_{p}.

Similarly, for i = 1,

exp(t_{1})N(x^{⊗n}, exp(t_{0}n))^{1/np} ⩽ exp(t_{1})k^{1/p} ⩽ ∥x∥_{p},

where the last inequality follows from the inequality of arithmetic and geometric means. Therefore, taking the limit n → ∞ in inequality (1) implies that ∥x∥ ⩽ exp(ε)∥x∥_{p}, and the result follows when ε goes to 0.

3. The case of L_{p} norms: proof of Theorem 1.2

Let ∥·∥ be a norm on the space E of simple random variables which satisfies the hypotheses of Theorem 1.2. Throughout the proof, we denote by B_{n} ∈ E a Bernoulli random variable with parameter 1/n, i.e. such that P(B_{n} = 1) = 1/n and P(B_{n} = 0) = 1 − 1/n. Moreover, we assume that the random variables (B_{n})_{n∈N} are independent.

We will define a norm ||| · ||| on c_{00} which will satisfy the hypotheses of Theorem 1.1. It is convenient to identify c_{00} with the union of an increasing sequence of subspaces

(2)  c_{00} = ⋃_{n∈N} R^{n}.

For x = (x_{1}, …, x_{n}) ∈ R^{n}, we define

|||x||| = \frac{∥X∥}{∥B_{n}∥},

where X ∈ E is a random variable with distribution \frac{1}{n}(δ_{x_{1}} + ⋯ + δ_{x_{n}}).

This defines a norm on c_{00} provided the construction is compatible with the union (2). To check this, consider x as an element of R^{m} for m > n, obtained by padding x with m − n zeros. Let X′ be a random variable with distribution \frac{1}{m}(δ_{x_{1}} + ⋯ + δ_{x_{n}} + (m − n)δ_{0}). If we moreover assume that the random variables X, X′, B_{n}, B_{m} are independent, it is easily checked that XB_{m} and X′B_{n} both have the distribution \frac{1}{nm}(δ_{x_{1}} + ⋯ + δ_{x_{n}}) + (1 − \frac{1}{nm})δ_{0}. By the hypotheses on the norm, this implies that ∥X∥ · ∥B_{m}∥ = ∥X′∥ · ∥B_{n}∥ and therefore

\frac{∥X∥}{∥B_{n}∥} = \frac{∥X′∥}{∥B_{m}∥}.

This shows that |||x||| is properly defined for x ∈ c_{00}. It is easily checked that ||| · ||| is a norm on c_{00} which is both permutation-invariant and multiplicative (for the latter, use the fact that B_{n}B_{m} and B_{nm} have the same distribution).

By Theorem 1.1, the norm ||| · ||| equals the norm of ℓ_{p} for some p ∈ [1, +∞]. To compute ∥B_{n}∥, consider the vector x ∈ R^{2n} given by n 1’s followed by n 0’s. We have

n^{1/p} = ∥x∥_{p} = |||x||| = \frac{∥B_{2}∥}{∥B_{2n}∥} = \frac{1}{∥B_{n}∥},

where the last equality follows from the aforementioned property of Bernoulli random variables. This implies that the equation

(3)  ∥X∥ = ∥X∥_{L_{p}}.

holds for every X ∈ E with rational weights, i.e. with distribution \frac{1}{n}(δ_{x_{1}} + ⋯ + δ_{x_{n}}) for some n. The extension to all random variables in E follows by an approximation argument. Indeed, for every positive random variable X ∈ E, there exist sequences (Y_{n}), (Z_{n}) of positive random variables, with rational weights, such that

Y_{n} ⩽ X ⩽ Z_{n}

and

lim_{n→∞} ∥Y_{n}∥_{L_{p}} = lim_{n→∞} ∥Z_{n}∥_{L_{p}} = ∥X∥_{L_{p}}.

Therefore, we may use the following lemma (a continuous version of lemma 2.1) to extend formula (3) to every X ∈ E.

Lemma 3.1. Let ∥ · ∥ be a norm on E which satisfies the hypotheses of Theorem 1.2. If X ∈ E, then the random variables X and |X| have the same norm. If X, Y ∈ E are two random variables such that 0 ⩽ X ⩽ Y, then ∥X∥ ⩽ ∥Y ∥.

Proof. To prove the first part, note that if ε is a random variable which is independent from X and such that P(ε = 1) = P(ε = −1) = 1/2, then εX and ε|X| are identically distributed. Assume now that 0 ⩽ X ⩽ Y. There exists a finite measurable partition (Ω_{1}, …, Ω_{n}) of Ω such that X and Y are constant on each set Ω_{i}. Let x_{i} (resp. y_{i}) be the value of X (resp. Y) on Ω_{i}; then x_{i} ⩽ y_{i}. For any ε = (ε_{1}, …, ε_{n}) ∈ {±1}^{n}, one may define a random variable Z_{ε} by setting Z_{ε}(ω) = ε_{i} for ω ∈ Ω_{i}. The random variable X can be written as a convex combination of the random variables {Z_{ε}Y }_{ε∈{±1}^{n}} (this is a consequence of the fact that (x_{1}, …, x_{n}) is in the convex hull of (±y_{1}, …, ±y_{n})—a fact already used in the proof of Lemma 2.1). We now conclude by the triangle inequality and the fact that ∥Z_{ε}Y ∥ = ∥Y ∥ since both variables are equal in absolute value. □

4. Extensions

4.1. Extension to the complex case. Theorems 1.1 and 1.2 extend easily to the complex case. We only state the discrete version. Up to a small detail, the proof is the same as in the real case.

Theorem 4.1. Let ∥ · ∥ be a permutation-invariant and multiplicative norm on the space of finitely supported complex sequences. Then, there exists some p ∈ [1, +∞] such that ∥·∥ = ∥·∥_{p}.

Proof. We argue in the same way as we did for real sequences. The proof adapts mutatis mutandis, except for the first part of Lemma 2.1 whose proof requires a slight modification. Let ω be a primitive k-th root of unity. If the coordinates of x and y differ only by a power of ω, then the vectors x ⊗ (1, ω, …, ω^{k−1}) and y ⊗ (1, ω, …, ω^{k−1}) are equal up to permutation of coordinates, and therefore ∥x∥ = ∥y∥. The case of a general complex phase follows by continuity. □

4.2. Noncommutative setting. Theorem 1.1 can be formulated to characterize the Schatten p-norms.

Let H be a infinite-dimensional (real or complex) separable Hilbert space and F (H) be the space of finite rank operators on H. Let ∥ · ∥ a norm on F (H) which is unitarily invariant: whenever U, V are unitary operators on H and A ∈ F (H), we have ∥UAV ∥ = ∥A∥. Assume also that the norm is multiplicative in the following sense: for any A, B ∈ F (H),

∥A ⊗ B∥ = ∥A∥ · ∥B∥.

As in the commutative case, we fix a isometry between H and the Hilbertian tensor product H ⊗ H to define ∥A ⊗ B∥—the particular choice we make is irrelevant because of the unitary invariance. The next theorem asserts that the only norms which are unitarily invariant and multiplicative are the Schatten p-norms defined as ∥A∥_{p} = (tr |A|^{p})^{1/p} for 1 ⩽ p < +∞, while p = ∞ corresponds to the operator norm.

Theorem 4.2. Let ∥ · ∥ be a norm on the space of finite-rank operators on a infinitedimensional Hilbert space which is both multiplicative and unitarily invariant. Then, there exists some p ∈ [1, +∞] such that ∥ · ∥ is the Schatten p-norm.

Proof. By a result of von Neumann (see [4], Theorem IV.2.1), a norm N on F (H) is unitarily invariant if and only if N(·) = ∥s(·)∥ for some symmetric norm ∥.∥ on c_{00}—here s(A) ∈ c_{00} denotes the list of singular values of an operator A ∈ F (H). The result follows then from the commutative case (Theorem 1.1 or Theorem 4.1). □

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Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, Batiment du Doyen Jean Braconnier, 43, boulevard du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France

  E-mail address: aubrun@math.univ-lyon1.fr

Laboratoire de Physique Théorique du CNRS, IRSAMC, Université de Toulouse, UPS, F-31062 Toulouse, France

  E-mail address: nechita@irsamc.ups-tlse.fr