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

Javier Alonso

Author Affiliations

Journal of Inequalities and Applications 2011, 2011:2 doi:10.1186/1029-242X-2011-2

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

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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

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

where x = (x1, x2). James [1] 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 [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 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 p - 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 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 [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) 2 be the coordinates of x in the basis . Let S = {x X : ||x|| = 1}, and for x S consider the linear functional fx: y X fx(y) = x2y1 - x1y2. Then it is immediate to see that fx 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.    □

### Competing interests

The author declares that they have no competing interests.

### Acknowledgements

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

### References

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)