##### Mathematical Sciences
Original research

# On some n-normed sequence spaces

Hemen Dutta1* and Hamid Mazaheri2

Author Affiliations

1 Department of Mathematics, Gauhati University, Guwahati, Assam 781014, India

2 Department of Mathematics, Yazd University, Yazd 89158, Iran

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Mathematical Sciences 2012, 6:56 doi:10.1186/2251-7456-6-56

 Received: 16 September 2012 Accepted: 9 October 2012 Published: 24 October 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we introduce the idea of constructing sequence spaces with elements in an n-norm space in comparison with the spaces c0, c, and the Orlicz space Mand extend the notion of n-norm to such spaces. Further we state and define some statements about the n-best approximation in n-normed spaces.

##### Keywords:
n-Norm; Locally convex space; n-orthogonality; Orlicz function; n-best approximation

### Introduction

The concept of 2-normed spaces was initially developed by Ghler [1] in the middle of 1960s, while that of n-normed spaces can be found in Misiak [2]. Since then, many others have studied this and related concepts and obtained various results; see for instance Lewandowska [3-5], Cho et al. [6], Gunawan [7,8], Gunawan and Mashadi [9], Dutta [10] and Esi [11-13].

Let nN and X be a real vector space of dimension d, where n ≤ d. A real-valued function ∥.,…,.,∥ on Xnsatisfying the following four conditions:

(1) ∥x1,x2,…,xn ∥ = 0 if and only if x1,x2,…,xnare linearly dependent,

(2) ∥x1,x2,…,xn∥ is invariant under permutation,

(3) ∥αx1,x2,…,xn∥=|α|∥x1,x2,…,xn∥ for any αR,

(4) ∥x + x,x2,…,xn∥≤∥x,x2,…,xn∥ + ∥x,x2,…,xn∥ is called an n-norm on X, and the pair (X,∥.,…,.∥) is called an n-normed space.

Let nN and X, a real vector space of dimension d, where 2 ≤ nd. βn − 1 be the collection of linearly independent sets B with n − 1 elements. For Bβn−1, let us define

Then pB is a seminorm on X and the family P = {pB: Bβn−1} of seminorms generates a locally convex topology on X.

Let (X,∥.,…,.∥) be an n-normed space and W1,W2,…,Wn be n subspaces of X. A map f:W1×W2×…×WnR is called an n-functional on W1×W2×…,Wn, whenever for all x11,x21,…,xn1W1, x12,x22,…,xn2W2,…,x1n,x2n,…,xnnWnand all λ1,λ2,…,λnR;(i) (ii) f(λ1x1,λ2x2,…,λnxn)=λ1λ2λnf(x1,x2,…,xn).

An n-functional f:W1×W2×…×WnRis called bounded if there exists a non-negative real number M (called a Lipschitz constant for f) such that |f(x1,x2,…,xn)|≤Mx1,x2,…,xn∥ for all x1W1,x2W2,…,xnWn. Also, the norm of an n-functional f is defined by

For an n-normed space (X,∥.,…,.∥) and 0≠u2,u3,…,unX, we denote by XB∗ the Banach space of all bounded n-functionals on X×<u2>×<u3>×…×<un>, where <z> be the subspace of X generated by z and B={u2,…,un}.

A sequence (xk) in an n-normed space (X,∥.,…,.∥) is said to converge to some LX in the n-norm if

A sequence (xk) in an n-normed space (X,∥.,…,.∥) is said to be Cauchy with respect to the n-norm if

If every Cauchy sequence in X converges to some LX, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.

Definition 2.1. Let (X,∥.,…,.∥) be an n-normed space. We say that x is n-orthogonal to y if ∥x,u2,u3,…,un∥≤∥x + αy,u2,u3,…,un∥, for all u2,u3,…,unX, αR and we write xny.

Definition 2.2. Let (X,∥.,…,.∥) be an n-normed space, M a nonempty subspace of X and xX, then g0M is called an n-best approximation of xX in M, if for every gM and u2,u3,…,unX,

If for every there exists at least one n-best approximation in M, then M is called n-proximinal subspace of X.

If for every there exists a unique n-best approximation in M, then M is called an n-Chebyshev subspace of X.

For xX we write,

Definition 2.3. A function M:[0,)→[0,), which is continuous, non-decreasing and convex with M(0) = 0, M(x) > 0, for x > 0 and M(x) → , as x is called an Orlicz function.

Let (X,∥.,…,.∥) be a real linear n-normed space and w (X) denotes X-valued sequence space. Then for an Orlicz function M, we define the following sequence spaces forsomeρ > 0, L and everyz2,…,znX:

and

When X=C, the complex field and M(x) = |x|, for all x ∈ [0,), the above spaces reduce to the spaces c, c0, and respectively.

It is obvious that

When L=0, we have (M,∥.,…,.∥)0=(M,∥.,…,.∥)1.

Lemma 2.1.The spaces (M,∥.,…,.∥)0, (M,∥.,…,.∥)1and (M,∥.,…,.∥)are linear spaces over the field of reals.

Proof. The proof is a routine verification and so omitted. □

### Methods

The ‘Introduction’ section recalls the notions of n-normed space, n-functional, Cauchy, and convergence sequences in n-normed spaces as well as defined the notions of n-orthogonality and n-best approximation and introduced three sequences spaces using an Orlicz function M with base space X, a real linear n-normed spaces in comparison with the classical spaces c0, c, and . In the ‘Results and discussion’ section, we prove some statements about the n-best approximation in n-normed spaces and investigate the introduced spaces for n-Banach spaces. The method applied for the main results is that first we give statement for each results and then each statement is supported with mathematical arguments as ‘proof’.

### Results and discussion

Now we state some statements about the n-best approximation in n-normed spaces and investigate the main results of this article involving the sequence spaces (M,∥.,…,.∥)0and (M,∥.,…,.∥)1 and (M,∥.,…,.∥).

Theorem 3.1.Let (X,∥.,…,.∥) be an n-normed linear space and 0 ≠ x, y ∈ X. Then the following statements are equivalent:

(i) xny.

(ii) There exist u2,…,unXandFXBsuch thatF∥ = 1, F(x,u2,…,un) = ∥x,u2,…,un∥, F(y,u2,…,un) = 0 andB = {u2,…,un}.

Corollary 3.2.Let (X,∥.,…,.∥) be an n-normed space, M a non-empty subspace of X, 0≠ xXand g0M. Then the following statements are equivalent: (i) g0PMn(x).(ii) There exist u2,…,unX and F ∈ X B ∗ such that ∥ F ∥ = 1, F (x − g0,u2,…,un) = ∥ x−g0,u2,…,unand F (g,u2,…,un) = 0 for all g ∈ M and B = {u2,…,un}.

Now we define an n-norm on the spaces (M,∥.,…,.∥)0 then (M,∥.,…,.∥)1 and (M,∥.,…,.∥) and prove that they are n-Banach spaces.

Lemma 3.1.Let Y be any one of the spaces (M,∥.,…,.∥)0then (M,∥.,…,.∥)1and (M,∥.,…,.∥). We define the following function (∥.,…,.∥)Yon Y × Y ×…× Y (n factors) by ∥ x1,…,xnY = 0 ifx1,…,xnare linearly dependent, and, if x1,…,xnare linearly independent.

Then ∥.,…,.∥Yis an n-norm on Y.

Proof. Proof is a routine verification and so omitted. □

Theorem 3.3.If X is an n-Banach space then the spaces (M,∥.,…,.∥)0and (M,∥.,…,.∥)1and (M,∥.,…,.∥)are n-Banach spaces.

Proof. Let Y be any one of the spaces (M,∥.,…,.∥)0and (M,∥.,…,.∥)1 and (M,∥.,…,.∥). Let (xi) be any Cauchy sequence in Y. Let x0 > 0 be fixed and t > 0 be such that for a 0 < ε < 1 and and x0t ≥ 1. Then there exists a positive integer n0 such that

Using the definition of n-norm, we get

Then for every z2,…,znX, we get

It follows that for every z2,…,znX,

For t>0 with , we have

Since an Orlicz function is non-decreasing, this implies that for every z2,…,znX,

Hence, (xi) is a Cauchy sequence in X for all kN and so convergent in X for all kN, since X is an n-Banach space. Suppose limixki=xk(say) for each . Now, using the continuity of Orlicz function M and n-norm, we can have

and as j . It follows that (xix)∈Y.

Since (xi)∈Y and Y is a linear space, so we have x=xi−(xix)∈Y. This completes the proof of the theorem.

Example 3.1. Consider the space C0 of real sequences with only finite number of non-zero terms. Let us define:

Then ∥.,…,.∥ is an n-norm on C0. That is not an n-norm on c0and lconsisting of real sequences.

### Conclusion

After observing the investigations of this paper, we can comment that while studying the n-normed structure, the main issue should be the use of the meaning of n-norms. We also observe that if a term in the definition of n-norm represents the change of shape and the n-norm stands for the associated area or center of gravity of the term, we can think of some plausible applicable of the notion of n-norm. As an example, we can think of the use of the notion of n-norm for a process where for a particular output we need n-inputs but with one main input and other (n-1)-inputs as dummy inputs to complete the process.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

HD wrote the abstract and background. Both authors wrote the preliminaries. Results concerning n-best approximation are proposed by HD and verified by HM. Results concerning n-normed spaces and n-Banach spaces are proposed by HM and verified by HD. Both authors read and approved the final manuscript.

### Acknowledgements

The authors thank the referee for the good recommendations.

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