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

Journal of Inequalities and Applications20112011:2

https://doi.org/10.1186/1029-242X-2011-2

• Received: 11 February 2011
• Accepted: 15 June 2011
• Published:

## Abstract

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

## Keywords

• Normed space
• Day-James space
• Birkhoff orthogonality

## 1991 Mathematics Subject Classification 46B20

where x = (x1, x2). James  considered the space p - 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. Birkhoff orthogonality coincides with usual orthogonality in inner product spaces. In arbitrary normed spaces Birkhoff orthogonality is in general not symmetric (e.g., in 2 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 , James [1, 3], and Day . The first reference related to the symmetry of Birkhoff orthogonality in two-dimensional spaces seems to be Radon  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  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 2 is called absolute if ||(x1, x2)|| = ||(|x1|, |x2|)|| for any (x1, x2) 2. Following Nilsrakoo and Saejung  let AN2 be the family of all absolute and normalized (i.e., ||(1, 0)|| = ||(0, 1)|| = 1) norms on 2. Examples of norms in AN 2 are p norms. Bonsall and Duncan  showed that there is a one-to-one correspondence between AN2 and the family Ψ2 of all continuous and convex functions ψ : [0, 1] 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  the family of norms ||·|| p,q of Day-James spaces p - q is extended to the family N2 of norms defined in 2 as

for ψ, φ Ψ2. The space 2 endowed with the above norm is called an ψ - φ space.

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

Lemma 1. 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) 2 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 (2, ||·|| 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 (2, ||·||2), where || (x1, x2) ||2 := ||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) ||2 = || (x1, x2) || ψ, φ for all (x1, x2) 2.    □

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
Department of Mathematics, University of Extremadura, 06006 Badajoz, Spain

## References 