1991 Mathematics Subject Classification 46B20

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 AN 2 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 ∈ Ψ2such 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.    □