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

- Javier Alonso
^{1}Email author

**2011**:2

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

© Alonso; licensee Springer. 2011

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

## Abstract

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.

## Keywords

- Normed space
- Day-James space
- Birkhoff orthogonality

## 1991 Mathematics Subject Classification 46B20

*ℓ*

_{ p }-

*ℓ*

_{ q }is defined for 1

*≤ p, q ≤ ∞*as the space ℝ

^{2}endowed with the norm

where *x* = (*x*_{1}, *x*_{2}). 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 ℝ^{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 [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.

*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.

^{2}is called absolute if ||(

*x*

_{1},

*x*

_{2})|| = ||(

*|x*

_{1}|, |

*x*

_{2}|)|| for any (

*x*

_{1},

*x*

_{2}) ∈ ℝ

^{2}. Following Nilsrakoo and Saejung [6] let

*AN*

_{2}be the family of all absolute and normalized (i.e., ||(1, 0)|| = ||(0, 1)|| = 1) norms on ℝ

^{2}. Examples of norms in

*AN*2 are

*ℓ*

_{ p }norms. Bonsall and Duncan [7] showed that there is a one-to-one correspondence between

*AN*

_{2}and the family Ψ

_{2}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

*AN*

_{2}, and by

for *ψ* in Ψ_{2}.

_{ p,q }of Day-James spaces

*ℓ*

_{ p }-

*ℓ*

_{ q }is extended to the family

*N*

_{2}of norms defined in ℝ

^{2}as

for *ψ, φ ∈* Ψ_{2}. The space ℝ_{2} 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 (*x*_{1}, *x*_{2}) *∈* ℝ^{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*) = *x*_{2}*y*_{1} - *x*_{1}*y*_{2}. Then it is immediate to see that *f*_{
x
} attains the norm in *y* ∈ *S* (i.e., |*x*_{2}*y*_{1} - *x*_{1}*y*_{2}| ≥ |*x*_{2}*z*_{1}*-x*_{1}*z*_{2}|, for all
) if and only if *y* ⊥ _{
B
} *x*. Therefore if *u, v* ∈ *S* are such that |*u*_{2}*v*_{1}*- u*_{1}*v*_{2}| = max_{(x, y)∈S×S}|*x*_{2}*y*_{1} - *x*_{1}*y*_{2}| then *u* ⊥ _{
B
} *v* and *v* ⊥ _{
B
} *u*. □

**Theorem 2** *For any two-dimensional normed space* (*X*, ||·||_{
X
}) *there exist ψ, φ ∈* Ψ_{2}*such that* (*X*, ||·||_{
X
}) *is isometrically isomorphic to* (ℝ^{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 (ℝ^{2}, ||·||_{ℝ2}), where || (*x*_{1}, *x*_{2}) ||_{ℝ2} := ||*x*_{1}*u* + *x*_{2}*v*||_{
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* ∈ Ψ_{2} and || (*x*_{1}, *x*_{2}) ||_{ℝ2} = || (*x*_{1}, *x*_{2}) || _{
ψ, φ
} for all (*x*_{1}, *x*_{2}) ∈ ℝ^{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

## References

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