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

## 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 $\parallel \left(x,y\right)\parallel =\parallel \left(|x|,|y|\right)\parallel$ for all $\left(x,y\right)\in {\mathbb{R}}^{2}$, and normalized if $\parallel \left(1,0\right)\parallel =\parallel \left(0,1\right)\parallel =1$. The set of all absolute normalized norms on ${\mathbb{R}}^{2}$ is denoted by $A{N}_{2}$. Bonsall and Duncan  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 $\left[0,1\right]$ satisfying $max\left\{1-t,t\right\}\le \psi \left(t\right)\le 1$ for all $t\in \left[0,1\right]$ (cf. ). The correspondence is given by the equation $\psi \left(t\right)=\parallel \left(1-t,t\right)\parallel$ for all $t\in \left[0,1\right]$. 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  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 , Day [5, 6] and James [3, 7, 8].

In 2006, Nilsrakoo and Saejung  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  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 $\left({\mathbb{R}}^{2},\parallel \cdot \parallel \right)$ 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 $\left({\mathbb{R}}^{2},\parallel \cdot \parallel \right)$ 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

$\phi \left(t\right)=\parallel \left(1-t,t\right)\parallel \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\psi \left(t\right)=\parallel \left(1-t,-t\right)\parallel$

for all $t\in \left[0,1\right]$, respectively. Then, clearly, we have $\phi ,\psi \in {\mathrm{\Psi }}_{2}$ and $\parallel \cdot \parallel ={\parallel \cdot \parallel }_{\phi ,\psi }$. Hence, the space $\left({\mathbb{R}}^{2},\parallel \cdot \parallel \right)$ 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 $i\in \left\{1,2,\dots ,n\right\}$. If $F:{X}^{n}\to \mathbb{R}$ is an n-linear form, then F is said to be alternating if

$F\left({x}_{1},\dots ,{x}_{i},{x}_{i+1},\dots ,{x}_{n}\right)=-F\left({x}_{1},\dots ,{x}_{i+1},{x}_{i},\dots ,{x}_{n}\right)$

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

$\parallel F\parallel :=sup\left\{|F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)|:\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\in {\left({S}_{X}\right)}^{n}\right\}<\mathrm{\infty },$

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

$|F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)|\le \parallel F\parallel \parallel {x}_{1}\parallel \parallel {x}_{2}\parallel \cdots \parallel {x}_{n}\parallel$

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

For our purpose, we give another simple proof of the following result of Day . For each subset A of a normed space, let $\left[A\right]$ 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 $\left[\left\{x\right\}\right]{\perp }_{B}M$.

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

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

$F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)=|\begin{array}{cccc}{\alpha }_{1}\left({x}_{1}\right)& {\alpha }_{2}\left({x}_{1}\right)& \dots & {\alpha }_{n}\left({x}_{1}\right)\\ {\alpha }_{1}\left({x}_{2}\right)& {\alpha }_{2}\left({x}_{2}\right)& \dots & {\alpha }_{n}\left({x}_{2}\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_{1}\left({x}_{n}\right)& {\alpha }_{2}\left({x}_{n}\right)& \dots & {\alpha }_{n}\left({x}_{n}\right)\end{array}|$

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

$F\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)=\parallel F\parallel >0.$

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

$\begin{array}{rcl}\parallel F\parallel \parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel & \ge & |F\left({e}_{1},\dots ,{e}_{i-1},\sum _{k=1}^{n}{\alpha }_{k}{e}_{k},{e}_{i+1},\dots ,{e}_{n}\right)|\\ =& |\sum _{k=1}^{n}{\alpha }_{k}F\left({e}_{1},\dots ,{e}_{i-1},{e}_{k},{e}_{i+1},\dots ,{e}_{n}\right)|\\ =& |{\alpha }_{i}F\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)|=\parallel F\parallel |{\alpha }_{i}|.\end{array}$

Thus, we obtain

$\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel \ge |{\alpha }_{i}|=\parallel {\alpha }_{i}{e}_{i}\parallel ,$

for all $i=1,2,\dots ,n$ and all $\left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\in {\mathbb{R}}^{n}$. This means that ${e}_{i}{\perp }_{B}\left[{\left\{{e}_{k}\right\}}_{k\ne i}\right]$ 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 $\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)$ of elements of ${S}_{X}$ such that ${e}_{i}{\perp }_{B}\left[{\left\{{e}_{k}\right\}}_{k\ne i}\right]$ for all $i=1,2,\dots ,n$. Since ${e}_{i}{\perp }_{B}\left[{\left\{{e}_{k}\right\}}_{k\ne i}\right]$, we have

$\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel \ge \parallel {\alpha }_{i}{e}_{i}\parallel =|{\alpha }_{i}|,$

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

$\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel \ge max\left\{|{\alpha }_{1}|,|{\alpha }_{2}|,\dots ,|{\alpha }_{n}|\right\}$

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

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

${\parallel \left({\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}\right)\parallel }_{0}=\parallel \sum _{k=1}^{n}{\alpha }_{k}{e}_{k}\parallel$

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

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

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## 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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Authors

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Correspondence to Kichi-Suke Saito.

### Competing interests

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 