On some n-normed sequence spaces

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, ℓ_∞and the Orlicz space ℓ_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.

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 n ∈ N 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,…,x_n ∥ = 0 if and only if x₁,x₂,…,x_nare linearly dependent,

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

(3) ∥αx₁,x₂,…,x_n∥=|α|∥x₁,x₂,…,x_n∥ for any α∈R,

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

Let n∈N and X, a real vector space of dimension d, where 2 ≤ n ≤ d. β_{n − 1} be the collection of linearly independent sets B with n − 1 elements. For B ∈ β_n−1, let us define

Then p_B is a seminorm on X and the family P = {p_B: B ∈ β_n−1} of seminorms generates a locally convex topology on X.

Let (X,∥.,…,.∥) be an n-normed space and W₁,W₂,…,W_n be n subspaces of X. A map f:W₁×W₂×…×W_n→R is called an n-functional on W₁×W₂×…,W_n, whenever for all x₁¹,x₂¹,…,x_n¹∈W₁, x₁²,x₂²,…,x_n²∈W₂,…,x₁ⁿ,x₂ⁿ,…,x_nⁿ∈W_nand all λ₁,λ₂,…,λ_n∈R;(i) (ii) f(λ₁x₁,λ₂x₂,…,λ_nx_n)=λ₁λ₂…λ_nf(x₁,x₂,…,x_n).

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

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

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

A sequence (x_k) 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 L∈X, 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,…,u_n∥≤∥x + αy,u₂,u₃,…,u_n∥, for all u₂,u₃,…,u_n ∈ X, α ∈ R and we write x ⊥ ⁿy.

Definition 2.2. Let (X,∥.,…,.∥) be an n-normed space, M a nonempty subspace of X and x ∈ X, then g₀ ∈ M is called an n-best approximation of x ∈ X in M, if for every g ∈ M and u₂,u₃,…,u_n ∈ X,

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 x ∈ X 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,…,z_n ∈ X:

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 ℓ_∞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. □

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 ℓ_∞. 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’.

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) x ⊥ⁿy.

(ii) There exist u₂,…,u_n∈XandF∈XB∗ such that ∥F∥ = 1, F(x,u₂,…,u_n) = ∥x,u₂,…,u_n∥, F(y,u₂,…,u_n) = 0 andB = {u₂,…,u_n}.

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

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,…,xⁿ∥_Y = 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,…,z_n∈X, we get

It follows that for every z₂,…,z_n∈X,

For t>0 with , we have

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

Hence, (xⁱ) is a Cauchy sequence in X for all k∈N and so convergent in X for all k∈N, since X is an n-Banach space. Suppose lim_i→∞x_kⁱ=x_k(say) for each . Now, using the continuity of Orlicz function M and n-norm, we can have

and as j→∞ . It follows that (xⁱ−x)∈Y.

Since (xⁱ)∈Y and Y is a linear space, so we have x=xⁱ−(xⁱ−x)∈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 l_∞consisting of real sequences.

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.

The authors declare that they have no competing interests.

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.

The authors thank the referee for the good recommendations.

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