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Every n-dimensional normed space is the space endowed with a normal norm
Journal of Inequalities and Applications volume 2012, Article number: 284 (2012)
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.
A norm on is said to be absolute if for all , and normalized if . The set of all absolute normalized norms on is denoted by . Bonsall and Duncan  showed the following characterization of absolute normalized norms on . Namely, the set of all absolute normalized norms on is in a one-to-one correspondence with the set of all convex functions ψ on satisfying for all (cf. ). The correspondence is given by the equation for all . Note that the norm associated with the function is given by
The Day-James space is defined for as the space endowed with the norm
James  considered the space with 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 , if for all . Birkhoff orthogonality is homogeneous, that is, implies for any real numbers α and β. However, Birkhoff orthogonality is not symmetric in general, that is, does not imply . 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 , where is defined for as the space 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 be a norm on . Then the space is a generalized Day-James space if and only if .
Proof If is a generalized Day-James space, then one can easily have . So, we assume that . Let
for all , respectively. Then, clearly, we have and . Hence, the space is a generalized Day-James space. □
Motivated by this fact, we consider the following
Definition 2 A norm on is said to be normal if it satisfies .
We recall some notions about multilinear forms. Let X be a real vector space. Then a real-valued function F on is said to be an n-linear form if it is linear separately in each variable, that is,
for each . If is an n-linear form, then F is said to be alternating if
for each or, equivalently, if for some i, j with . Furthermore, F is said to be bounded if
where denotes the unit sphere of X. If F is bounded, then we have
for all .
For our purpose, we give another simple proof of the following result of Day . For each subset A of a normed space, let 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 , if for all and all . In particular, denotes .
Lemma 3 Let X be an n-dimensional real normed space. Then there exists a basis for X such that and for all .
Proof Let be a basis for X. Then each vector is uniquely expressed in the form . Define the function F on by
for all . Then it is easy to check that F is an alternating bounded n-linear form. Since F is jointly continuous on the compact subset of , there exists such that
For all and all , we have
Thus, we obtain
for all and all . This means that for all . □
Now, we state the main theorem.
Theorem 4 Every n-dimensional normed space is isometrically isomorphic to the space endowed with a normal norm.
Proof By Lemma 3, there exists an n-tuple of elements of such that for all . Since , we have
for all and all . Hence, we obtain
for all . From this fact, we note that is linearly independent, that is, a basis for X.
Define the norm on by the formula
for all . Then, clearly, is normal and X is isometrically isomorphic to the space . This completes the proof. □
Since the space 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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The second author was supported in part by Grants-in-Aid for Scientific Research (No. 23540189), Japan Society for the Promotion of Science.
The authors declare that they have no competing interests.
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 endowed with a normal norm. J Inequal Appl 2012, 284 (2012). https://doi.org/10.1186/1029-242X-2012-284
- normed space
- Day-James space
- Birkhoff orthogonality