Any two-dimensional Normed space is a generalized Day-James space

It is proved that any two-dimensional normed space is isometrically isomorphic to a generalized Day-James space ℓ_ψ-ℓ_φ, introduced by W. Nilsrakoo and S. Saejung.

The Day-James space ℓ_p-ℓ_q is defined for 1 ≤ p, q ≤ ∞ as the space ℝ² endowed with the norm

where x = (x₁, x₂). James [1] considered the space ℓ_p - ℓ_q with 1/p + 1/q = 1 as an example of a two-dimensional normed space where Birkhoff orthogonality is symmetric. Recall that if x and y are vectors in a normed space then x is said to be Birkhoff orthogonal to y, (x ⊥_B y), if ||x +λy|| ≥||x|| for every scalar λ [2]. Birkhoff orthogonality coincides with usual orthogonality in inner product spaces. In arbitrary normed spaces Birkhoff orthogonality is in general not symmetric (e.g., in ℝ² with ||·||_∞), and it is symmetric in a normed space of three or more dimension if and only if the norm is induced by an inner product. This last significant property was obtained in gradual stages by Birkhoff [2], James [1,3], and Day [4]. The first reference related to the symmetry of Birkhoff orthogonality in two-dimensional spaces seems to be Radon [5] in 1916. He considered plane convex curves with conjugate diameters (as in ellipses) in order to solve certain variational problems.

The procedure that James used to get two-dimensional normed spaces where Birkhoff orthogonality is symmetric was extended by Day [4] in the following way. Let (X, ||·||_X) be a two-dimensional normed space and let u, v ∈ X be such that ||u||_X = ||v||_X = 1, u ⊥_B v, and v ⊥_B u (see Lemma below). Then, taking a coordinate system where u = (1, 0) and v = (0, 1) and defining

one gets that in the space (X, ||·||_X,X*) Birkhoff orthogonality is symmetric. Moreover, Day also proved that surprisingly the norm of any two-dimensional space where Birkhoff orthogonality is symmetric can be constructed in the above way.

A norm on ℝ² is called absolute if ||(x₁, x₂)|| = ||(|x₁|, |x₂|)|| for any (x₁, x₂) ∈ ℝ². Following Nilsrakoo and Saejung [6] let AN₂ be the family of all absolute and normalized (i.e., ||(1, 0)|| = ||(0, 1)|| = 1) norms on ℝ². Examples of norms in AN2 are ℓ_p norms. Bonsall and Duncan [7] showed that there is a one-to-one correspondence between AN₂ and the family Ψ₂ of all continuous and convex functions ψ : [0, 1] → ℝ such that ψ(0) = ψ(1) = 1 and max{1-t, t} ≤ ψ(t) ≤ 1 (0 ≤ t ≤ 1). The correspondence is given by ψ(t) = ||(1-t, t)|| for ||·|| in AN₂, and by

for ψ in Ψ₂.

In [6] the family of norms ||·||_p,q of Day-James spaces ℓ_p - ℓ_q is extended to the family N₂ of norms defined in ℝ² as

for ψ, φ ∈ Ψ₂. The space ℝ₂ endowed with the above norm is called an ℓ_ψ-ℓ_φ space.

The purpose of this paper is to show that any two-dimensional normed space is isometrically isomorphic to an ℓ_ψ-ℓ_φ space. To this end we shall use the following lemma due to Day [8]. The nice proof we reproduce here is taken from the PhD Thesis of del Río [9], and is based on explicitly developing the idea underlying one of the two proofs given by Day.

Lemma 1 [8]. Let (X, ||·||) be a two-dimensional normed space. Then, there exist u, v ∈ X such that ||u|| = ||v|| = 1, u ⊥_B v, and v ⊥_B u.

Proof. Let e, be linearly independent, and for x ∈ X let (x₁, x₂) ∈ ℝ² be the coordinates of x in the basis . Let S = {x ∈ X : ||x|| = 1}, and for x ∈ S consider the linear functional f_x: y ∈ X ↦ f_x(y) = x₂y₁ - x₁y₂. Then it is immediate to see that f_x attains the norm in y ∈ S (i.e., |x₂y₁ - x₁y₂| ≥ |x₂z₁ -x₁z₂|, for all ) if and only if y ⊥_B x. Therefore if u, v ∈ S are such that |u₂v₁ - u₁v₂| = max_{(x, y)∈S×S} |x₂y₁ - x₁y₂| then u ⊥_B v and v ⊥_B u.    □

Theorem 2 For any two-dimensional normed space (X, ||·||_X) there exist ψ, φ ∈ Ψ₂ such that (X, ||·||_X) is isometrically isomorphic to (ℝ², ||·||_{ψ, φ}).

Proof. By Lemma 1 we can take u, v ∈ X such that ||u|| = ||v|| = 1, u ⊥_B v, and v ⊥_B u. Then u and v are linearly independent and (X, ||·||_X) is isometrically isomorphic to (ℝ², ||·||_ℝ2), where || (x₁, x₂) ||_ℝ2 := ||x₁u + x₂v||_X. Defining ψ(t) = || (1 - t)u + tv||_X, φ(t) = || (1 - t)u - tv||_X, (0 ≤ t ≤ 1), one trivially has that ψ, φ ∈ Ψ₂ and || (x₁, x₂) ||_ℝ2 = || (x₁, x₂) ||_{ψ, φ} for all (x₁, x₂) ∈ ℝ².    □

The author declares that they have no competing interests.

Research partially supported by MICINN (Spain) and FEDER (UE) grant MTM2008-05460, and by Junta de Extremadura grant GR10060 (partially financed with FEDER).

1. James, RC: Inner products in normed linear spcaces. Bull Am Math Soc. 53, 559–566 (1947). Publisher Full Text

2. Birkhoff, G: Orthogonality in linear metric spaces. Duke Math J. 1, 169–172 (1935). Publisher Full Text

3. James, RC: Orthogonality and linear functionals in normed linear spaces. Trans Am Math Soc. 61, 265–292 (1947). Publisher Full Text

4. Day, MM: Some characterizations of inner product spaces. Trans Am Math Soc. 62, 320–337 (1947). Publisher Full Text

5. Radon, J: Über eine besondere Art ebener konvexer Kurven. Leipziger Berichre, Math Phys Klasse. 68, 23–28 (1916)

6. Nilsrakoo, W, Saejung, S: The James constant of normalized norms on ℝ². J Ineq Appl. 2006, 1–12 Article ID 26265 (2006)

7. Bonsall, FF, Duncan, J: Numerical ranges II. Lecture Note Series in London Mathematical Society. Cambridge University Press, Cambridge (1973)

8. Day, MM: Polygons circumscribed about closed convex curves. Trans Am Math Soc. 62, 315–319 (1947). Publisher Full Text

9. del Río, M: Ortogonalidad en Espacios Normados y Caracterización de Espacios Prehilbertianos. Dpto. de Análisis Matemático, Univ. de Santiago de Compostela, Spain, Serie B. 14, (1975)