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# Every *n*-dimensional normed space is the space {\mathbb{R}}^{n} endowed with a normal norm

*Journal of Inequalities and Applications*
**volume 2012**, Article number: 284 (2012)

## Abstract

Recently, Alonso showed that every two-dimensional normed space is isometrically isomorphic to a generalized Day-James space introduced by Nilsrakoo and Saejung. In this paper, we consider the result of Alonso for *n*-dimensional normed spaces.

**MSC:**46B20.

A norm \parallel \cdot \parallel on {\mathbb{R}}^{2} is said to be absolute if \parallel (x,y)\parallel =\parallel (|x|,|y|)\parallel for all (x,y)\in {\mathbb{R}}^{2}, and normalized if \parallel (1,0)\parallel =\parallel (0,1)\parallel =1. The set of all absolute normalized norms on {\mathbb{R}}^{2} is denoted by A{N}_{2}. Bonsall and Duncan [1] showed the following characterization of absolute normalized norms on {\mathbb{R}}^{2}. Namely, the set A{N}_{2} of all absolute normalized norms on {\mathbb{R}}^{2} is in a one-to-one correspondence with the set {\mathrm{\Psi}}_{2} of all convex functions *ψ* on [0,1] satisfying max\{1-t,t\}\le \psi (t)\le 1 for all t\in [0,1] (*cf.* [2]). The correspondence is given by the equation \psi (t)=\parallel (1-t,t)\parallel for all t\in [0,1]. Note that the norm {\parallel \cdot \parallel}_{\psi} associated with the function \psi \in {\mathrm{\Psi}}_{2} is given by

The Day-James space {\ell}_{p}\text{-}{\ell}_{q} is defined for 1\le p,q\le \mathrm{\infty} as the space {\mathbb{R}}^{2} endowed with the norm

James [3] considered the space {\ell}_{p}\text{-}{\ell}_{q} with {p}^{-1}+{q}^{-1}=1 as an example of a two-dimensional normed space where Birkhoff orthogonality is symmetric. Recall that if *x*, *y* are elements of a real normed space *X*, then *x* is said to be Birkhoff-orthogonal to *y*, denoted by x{\perp}_{B}y, if \parallel x+\lambda y\parallel \ge \parallel x\parallel for all \lambda \in \mathbb{R}. Birkhoff orthogonality is *homogeneous*, that is, x{\perp}_{B}y implies \alpha x{\perp}_{B}\beta y for any real numbers *α* and *β*. However, Birkhoff orthogonality is not *symmetric* in general, that is, x{\perp}_{B}y does not imply y{\perp}_{B}x. More details about Birkhoff orthogonality can be found in Birkhoff [4], Day [5, 6] and James [3, 7, 8].

In 2006, Nilsrakoo and Saejung [9] introduced and studied generalized Day-James spaces {\ell}_{\phi}\text{-}{\ell}_{\psi}, where {\ell}_{\phi}\text{-}{\ell}_{\psi} is defined for \phi ,\psi \in {\mathrm{\Psi}}_{2} as the space {\mathbb{R}}^{2} endowed with the norm

Recently, Alonso [10] showed that every two-dimensional normed space is isometrically isomorphic to a generalized Day-James space. In this paper, we consider the result of Alonso for *n*-dimensional spaces.

First, we give a characterization of generalized Day-James spaces.

**Proposition 1** *Let* \parallel \cdot \parallel *be a norm on* {\mathbb{R}}^{2}. *Then the space* ({\mathbb{R}}^{2},\parallel \cdot \parallel ) *is a generalized Day*-*James space if and only if* {\parallel \cdot \parallel}_{\mathrm{\infty}}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel}_{1}.

*Proof* If ({\mathbb{R}}^{2},\parallel \cdot \parallel ) is a generalized Day-James space, then one can easily have {\parallel \cdot \parallel}_{\mathrm{\infty}}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel}_{1}. So, we assume that {\parallel \cdot \parallel}_{\mathrm{\infty}}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel}_{1}. Let

for all t\in [0,1], respectively. Then, clearly, we have \phi ,\psi \in {\mathrm{\Psi}}_{2} and \parallel \cdot \parallel ={\parallel \cdot \parallel}_{\phi ,\psi}. Hence, the space ({\mathbb{R}}^{2},\parallel \cdot \parallel ) is a generalized Day-James space. □

Motivated by this fact, we consider the following

**Definition 2** A norm \parallel \cdot \parallel on {\mathbb{R}}^{n} is said to be normal if it satisfies {\parallel \cdot \parallel}_{\mathrm{\infty}}\le \parallel \cdot \parallel \le {\parallel \cdot \parallel}_{1}.

We recall some notions about multilinear forms. Let *X* be a real vector space. Then a real-valued function *F* on {X}^{n} is said to be an *n*-linear form if it is linear separately in each variable, that is,

for each i\in \{1,2,\dots ,n\}. If F:{X}^{n}\to \mathbb{R} is an *n*-linear form, then *F* is said to be alternating if

for each i\in \{1,2,\dots ,n\} or, equivalently, F({x}_{1},{x}_{2},\dots ,{x}_{n})=0 if {x}_{i}={x}_{j} for some *i*, *j* with i\ne j. Furthermore, *F* is said to be bounded if

where {S}_{X} denotes the unit sphere of *X*. If *F* is bounded, then we have

for all ({x}_{1},{x}_{2},\dots ,{x}_{n})\in {X}^{n}.

For our purpose, we give another simple proof of the following result of Day [5]. For each subset *A* of a normed space, let [A] denote the closed linear span of *A*. If *M*, *N* are subspaces of a real normed space *X*, then *M* is said to be Birkhoff orthogonal to *N*, denoted by M{\perp}_{B}N, if \parallel x+y\parallel \ge \parallel x\parallel for all x\in M and all y\in N. In particular, x{\perp}_{B}M denotes [\{x\}]{\perp}_{B}M.

**Lemma 3** *Let* *X* *be an* *n*-*dimensional real normed space*. *Then there exists a basis* \{{e}_{1},{e}_{2},\dots ,{e}_{n}\} *for* *X* *such that* \parallel {e}_{i}\parallel =1 *and* {e}_{i}{\perp}_{B}[{\{{e}_{k}\}}_{k\ne i}] *for all* i=1,2,\dots ,n.

*Proof* Let \{{u}_{1},{u}_{2},\dots ,{u}_{n}\} be a basis for *X*. Then each vector x\in X is uniquely expressed in the form x={\sum}_{k=1}^{n}{\alpha}_{k}(x){u}_{k}. Define the function *F* on {X}^{n} by

for all ({x}_{1},{x}_{2},\dots ,{x}_{n})\in {X}^{n}. Then it is easy to check that *F* is an alternating bounded *n*-linear form. Since *F* is jointly continuous on the compact subset {({S}_{X})}^{n} of {X}^{n}, there exists ({e}_{1},{e}_{2},\dots ,{e}_{n})\in {({S}_{X})}^{n} such that

For all i=1,2,\dots ,n and all ({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})\in {\mathbb{R}}^{n}, we have

Thus, we obtain

for all i=1,2,\dots ,n and all ({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})\in {\mathbb{R}}^{n}. This means that {e}_{i}{\perp}_{B}[{\{{e}_{k}\}}_{k\ne i}] for all i=1,2,\dots ,n. □

Now, we state the main theorem.

**Theorem 4** *Every* *n*-*dimensional normed space is isometrically isomorphic to the space* {\mathbb{R}}^{n} *endowed with a normal norm*.

*Proof* By Lemma 3, there exists an *n*-tuple ({e}_{1},{e}_{2},\dots ,{e}_{n}) of elements of {S}_{X} such that {e}_{i}{\perp}_{B}[{\{{e}_{k}\}}_{k\ne i}] for all i=1,2,\dots ,n. Since {e}_{i}{\perp}_{B}[{\{{e}_{k}\}}_{k\ne i}], we have

for all i=1,2,\dots ,n and all ({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})\in {\mathbb{R}}^{n}. Hence, we obtain

for all ({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})\in {\mathbb{R}}^{n}. From this fact, we note that \{{e}_{1},{e}_{2},\dots ,{e}_{n}\} is linearly independent, that is, a basis for *X*.

Define the norm {\parallel \cdot \parallel}_{0} on {\mathbb{R}}^{n} by the formula

for all ({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})\in {\mathbb{R}}^{n}. Then, clearly, {\parallel \cdot \parallel}_{0} is normal and *X* is isometrically isomorphic to the space ({\mathbb{R}}^{n},{\parallel \cdot \parallel}_{0}). This completes the proof. □

Since the space {\mathbb{R}}^{2} endowed with a normal norm is a generalized Day-James space by Proposition 1, we have the result of Alonso as a corollary.

**Corollary 5** ([10])

*Every two*-*dimensional real normed space is isometrically isomorphic to a generalized Day*-*James space*.

## References

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*Numerical Ranges II*. Cambridge University Press, Cambridge; 1973.Saito KS, Kato M, Takahashi Y:Von Neumann-Jordan constant of absolute normalized norms on {\mathbb{C}}^{2}.

*J. Math. Anal. Appl.*2000, 244: 515–532. 10.1006/jmaa.2000.6727James RC: Inner products in normed linear spaces.

*Bull. Am. Math. Soc.*1947, 53: 559–566. 10.1090/S0002-9904-1947-08831-5Birkhoff G: Orthogonality in linear metric spaces.

*Duke Math. J.*1935, 1: 169–172. 10.1215/S0012-7094-35-00115-6Day MM: Polygons circumscribed about closed convex curves.

*Trans. Am. Math. Soc.*1947, 62: 315–319. 10.1090/S0002-9947-1947-0022686-9Day MM: Some characterizations of inner product spaces.

*Trans. Am. Math. Soc.*1947, 62: 320–337. 10.1090/S0002-9947-1947-0022312-9James RC: Orthogonality in normed linear spaces.

*Duke Math. J.*1945, 12: 291–302. 10.1215/S0012-7094-45-01223-3James RC: Orthogonality and linear functionals in normed linear spaces.

*Trans. Am. Math. Soc.*1947, 61: 265–292. 10.1090/S0002-9947-1947-0021241-4Nilsrakoo W, Saejung S:The James constant of normalized norms on {\mathbb{R}}^{2}.

*J. Inequal. Appl.*2006., 2006: Article ID 26265Alonso J: Any two-dimensional normed space is a generalized Day-James space.

*J. Inequal. Appl.*2011., 2011: Article ID 2

## Acknowledgements

The second author was supported in part by Grants-in-Aid for Scientific Research (No. 23540189), Japan Society for the Promotion of Science.

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The authors declare that they have no competing interests.

### Authors’ contributions

RT conceived of the study, carried out the study of a structure of finite dimensional normed linear spaces, and drafted the manuscript. KS participated in the design of the study and helped to draft the manuscript. All authors read and approved the final manuscripts.

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Tanaka, R., Saito, KS. Every *n*-dimensional normed space is the space {\mathbb{R}}^{n} endowed with a normal norm.
*J Inequal Appl* **2012, **284 (2012). https://doi.org/10.1186/1029-242X-2012-284

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DOI: https://doi.org/10.1186/1029-242X-2012-284

### Keywords

- normed space
- Day-James space
- Birkhoff orthogonality