ON n-NORMED SPACES

Given ann-normed space withn≥ 2, we offer a simple way to derive an (n−1)norm from the n-norm and realize that any n-normed space is an (n−1)-normed space. We also show that, in certain cases, the (n−1)-norm can be derived from the n-norm in such a way that the convergence and completeness in the n-norm is equivalent to those in the derived (n− 1)-norm. Using this fact, we prove a fixed point theorem for some n-Banach spaces. 2000 Mathematics Subject Classification. 46B20, 46B99, 46A19, 46A99, 47H10.

The theory of 2-normed spaces was first developed by Gähler [3] in the mid 1960's, while that of n-normed spaces can be found in [11]. Recent results can be found, for example, in [9,10]. Related works on n-metric spaces and n-inner product spaces may be found, for example, in [1,2,4,5,7,6,12].
In this note, we will show that every n-normed space with n ≥ 2 is an (n−1)-normed space and hence, by induction, an (n − r )-normed space for all r = 1,...,n − 1. In particular, given an n-normed space, we offer a simple way to derive an (n − 1)-norm from the n-norm, different from that in [5].
We will also apply our result to study convergence and completeness in n-normed spaces, which will be defined later. This enables us to prove a fixed point theorem for some n-normed spaces.
The case n = 2 was previously studied in [8].
Corollary 2.2. Every n-normed space is an (n − r )-normed space for all r = 1,...,n− 1. In particular, every n-normed space is a normed space. norms, however, are equivalent to ·,...,· ∞ , as long as we use the same set of n vectors a 1 ,...,a n . In certain cases, it is possible to get equivalent (n − 1)-norms even if we use different sets of n vectors.

The standard case.
Take a look at a standard example. Let X be a real inner product space of dimension d ≥ n. Equip X with the standard n-norm x 1 ,...,x n S := where ·, · denotes the inner product on X. (If X = R n , then this n-norm is exactly the same as the Euclidean n-norm ·,...,· E mentioned earlier.) Notice that for n = 1, the above n-norm is the usual norm gives the length of x 1 , while for n = 2, it defines the standard 2-norm x 3 E is nothing but the volume of the parallelograms spanned by x 1 , x 2 , and x 3 . In general, x 1 ,...,x n S represents the volume of the n-dimensional parallelepiped spanned by x 1 ,...,x n in X. Now let {e 1 ,...,e n } be an orthonormal set in X. Then, by Theorem 2.1, the following function defines an (n − 1)-norm on X. Further, we have the following fact.
Next, take a unit vector e = α 1 e 1 +···+α n e n such that e ⊥ span{x 1 ,...,x n−1 }. (Here we are still assuming that x 1 ,...,x n−1 are linearly independent.) Then, by properties (3) and (4) of the n-norm, we have But, by the Cauchy-Schwarz inequality, we have Hence we obtain 10) and this completes the proof.
As we will see later, we can obtain a better (n − 1)-norm by using a set of d, rather than just n, linearly independent vectors in X (that is, by using a basis for X).

Applications and further results.
Recall that a sequence x(k) in an n-normed space (X, ·,...,· ) is said to converge to an x ∈ X (in the n-norm) whenever for every x 1 ,...,x n−1 ∈ X.
The following proposition says that the convergence in the n-norm implies the convergence in the derived (n − 1)-norm ·,...,· ∞ , defined with respect to an arbitrary linearly independent set {a 1 ,...,a n } in X. Proposition 3.1. If x(k) converges to an x ∈ X in the n-norm, then x(k) also converges to x in the derived (n − 1)-norm ·,...,· ∞ , that is, for every x 1 ,...,x n−2 ∈ X.

The standard case.
In a standard n-normed space (X, ·,...,· S ), the converse of Proposition 3.1 is also true, especially when the derived (n − 1)-norm ·,...,· ∞ is defined with respect to an orthonormal set {e 1 ,...,e n } in X as in Section 2.1. Proof. Suppose that x(k) converges to an x ∈ X in the derived (n − 1)-norm ·,...,· ∞ . We want to show that x(k) also converges to x in the n-norm. Take x 1 ,...,x n−1 ∈ X. Then one may observe that where ·,...,· S and · S on the right-hand side denote the standard (n − 1)-norm and the usual norm on X, respectively. By Fact 2.4, we have that is, x(k) converges to x in the n-norm. Then, as mentioned before, the function ·,...,· defines an (n − 1)-norm on X. With this derived (n − 1)-norm, we have the following result. Proof. If a sequence in X is convergent in the n-norm, then it will certainly be convergent in the (n − 1)-norm ·,...,· . Conversely, suppose that x(k) converges to an x ∈ X in ·,...,· . Take x 1 ,...,x n−1 ∈ X. Writing x n−1 = α 1 b 1 + ··· + α d b d , we get that is, x(k) converges to x in the n-norm.

The standard, separable case.
We go back to the standard case, where X is a real inner product space of dimension d ≥ n equipped with the standard n-norm ·,...,· S as in Section 2.1. But suppose now that X is separable and that {e i : i ∈ I d }, , is an orthonormal basis for X. For every x 1 ,...,x n−1 ∈ X and every basis vector e i (i ∈ I d ), we have where ·,...,· S on the right-hand side denotes the standard (n − 1)-norm on X. Hence, with respect to {e i : i ∈ I d }, we may define the function ·,...,· on X n−1 by x 1 ,...,x n−1 := sup x 1 ,...,x n−1 ,e i S : i ∈ I d (3.12) and check that it also defines an (n − 1)-norm on X. Moreover, we have the following relation between the two derived (n − 1)-norms ·,...,· and ·,...,· ∞ (the latter being defined with respect to {e 1 ,...,e n } only): for every x 1 ,...,x n−1 ∈ X. Hence we conclude the following fact.
Fact 3.5. On a standard n-normed space X, the two derived (n−1)-norms ·,...,· ∞ and ·,...,· and the standard (n − 1)-norm ·,...,· S are equivalent. Accordingly, a sequence in a standard n-normed space X is convergent in the n-norm if and only if it is convergent in one of the three (n − 1)-norms.

Cauchy sequences, completeness and fixed point theorem.
Recall that a sequence x(k) in an n-normed space (X, ·,...,· ) is called Cauchy (with respect to the n-norm) if lim k,l→∞ x 1 ,...,x n−1 , x(k) − x(l) = 0 (3.14) for every x 1 ,...,x n−1 ∈ X. If every Cauchy sequence in X converges to an x ∈ X, then X is said to be complete (with respect to the n-norm). A complete n-normed space is then called an n-Banach space. By replacing the phrases "x(k) converges to x" with "x(k) is Cauchy" and "x(k)−x" with "x(k)−x(l)," we see that the analogues of Proposition 3.1, Fact 3.2, Corollary 3.3, Proposition 3.4, and Fact 3.5 hold for Cauchy sequences.
Hence, for the standard or finite-dimensional case, we have the following result. Consequently, we have the following result.
Corollary 3.7 (fixed point theorem). Let (X, ·,...,· ) be a standard or finitedimensional n-Banach space, and T a contractive mapping of X into itself, that is, there exists a constant C ∈ (0, 1) such that for all x 1 ,...,x n−1 ,y,z in X. Then T has a unique fixed point in X.
Proof. First consider the case n = 2 (see [8]). By Proposition 3.6, we know that X is a Banach space with respect to the derived norm · ∞ (for standard case) or · (for finite-dimensional case). Since the mapping T is also contractive with respect to · ∞ or · , we conclude by the fixed point theorem for Banach spaces that T has a unique fixed point in X. For n > 2, the result follows by induction.
Remark 3.8. In the finite-dimensional case, it is actually enough to assume that X is an n-normed space because we know that all finite-dimensional normed spaces are complete and, by Proposition 3.6(b), so are all finite-dimensional n-normed spaces.

Concluding remark.
We have shown that an n-normed space with n ≥ 2 is an (n − 1)-normed space and that, for the standard or finite-dimensional case, the (n−1)-norm can be derived from the n-norm in such a way that the convergence and completeness in the n-norm is equivalent to those in the derived (n − 1)-norm.
Below is an example of a non-standard, infinite-dimensional 2-normed space for which we can derive a norm from the 2-norm such that the convergence and completeness in the 2-norm is equivalent to those in the derived norm.
Let X = l ∞ , the space of bounded sequences of real numbers. Equip X with the following 2-norm With respect to {a 1 ,a 2 }, we derive the norm · ∞ via x ∞ := max x, a 1 , x, a 2 . (4.2) But x, a 1 = sup i∈N\{1} |x i | and x, a 2 = sup i∈N\{2} |x i |, and so we obtain the usual norm on l ∞ . Now suppose that x(k) is a sequence in X that converges to x in the derived norm · ∞ . For every y ∈ X, we have Thus, for this particular example, we see that the convergence in the 2-norm is equivalent to that in the derived norm. By similar arguments, we can also verify that the completeness in the 2-norm is equivalent to that in the derived norm.
For general non-standard, infinite-dimensional n-normed spaces, however, it is unknown whether we can always derive an (n − 1)-norm from the n-norm such that the convergence and completeness in the n-norm is equivalent to those in the derived (n − 1)-norm.