In the present paper we introduce some sequence spaces over n-normed spaces defined by a Musielak-Orlicz function ℳ = (M_k). We also study some topological properties and prove some inclusion relations between these spaces.

1. Introduction and Preliminaries

An Orlicz function M is a function, which is continuous, nondecreasing, and convex with M(0) = 0, M(x) > 0 for x > 0 and M(x) → ∞ as x → ∞.

Lindenstrauss and Tzafriri [1] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x = (x_k); then

()

which is called as an Orlicz sequence space. The space ℓ_M is a Banach space with the norm

()

It is shown in [1] that every Orlicz sequence space ℓ_M contains a subspace isomorphic to ℓ_p (p ≥ 1). The Δ₂-condition is equivalent to M(Lx) ≤ kLM(x) for all values of x ≥ 0 and for L > 1. A sequence ℳ = (M_k) of Orlicz functions is called a Musielak-Orlicz function (see [2, 3]). A sequence 𝒩 = (N_k) defined by

()

is called the complementary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space t_ℳ and its subspace h_ℳ are defined as follows:

()

where I_ℳ is a convex modular defined by

()

We consider t_ℳ equipped with the Luxemburg norm

()

or equipped with the Orlicz norm

()

Let X be a linear metric space. A function p : X→ℝ is called paranorm if

(1)
p(x) ≥ 0 for all x ∈ X,
(2)
p(−x) = p(x) for all x ∈ X,
(3)
p(x + y) ≤ p(x) + p(y) for all x, y ∈ X,
(4)
(λ_n) is a sequence of scalars with λ_n → λ as n → ∞ and (x_n) is a sequence of vectors with p(x_n − x) → 0 as n → ∞; then p(λ_nx_n − λx) → 0 as n → ∞.

A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [4], Theorem 10.4.2, pp. 183). For more details about sequence spaces, see [5–12] and references therein.

A sequence of positive integers θ = (k_r) is called lacunary if k₀ = 0, 0 < k_r < k_r+1 and h_r = k_r − k_r−1 → ∞ as r → ∞. The intervals determined by θ will be denoted by I_r = (k_r−1, k_r) and q_r = k_r/k_r−1. The space of lacunary strongly convergent sequences N_θ was defined by Freedman et al. [13] as

()

Strongly almost convergent sequence was introduced and studied by Maddox [14] and Freedman et al. [13]. Parashar and Choudhary [15] have introduced and examined some properties of four sequence spaces defined by using an Orlicz function M, which generalized the well-known Orlicz sequence spaces [C, 1, p], [C, 1, p] ₀, and [C, 1, p] _∞. It may be noted here that the space of strongly summable sequences was discussed by Maddox [16] and recently in [17].

Mursaleen and Noman [18] introduced the notion of λ-convergent and λ-bounded sequences as follows.

Let

be a strictly increasing sequence of positive real numbers tending to infinity; that is,

()

and it is said that a sequence x = (x_k) ∈ w is λ-convergent to the number L, called the λ-limit of x if Λ_m(x) → L as m → ∞, where

()

The sequence x = (x_k) ∈ w is λ-bounded if sup _m | Λ_m(x)| < ∞. It is well known [18] that if lim _m x_m = a in the ordinary sense of convergence, then

()

This implies that

()

which yields that lim _m Λ_m (x) = a and hence x = (x_k) ∈ w is λ-convergent to a.

The concept of 2-normed spaces was initially developed by Gähler [19] in the mid 1960s, while for that of n-normed spaces one can see Misiak [20]. Since then, many others have studied this concept and obtained various results; see Gunawan ([21, 22]) and Gunawan and Mashadi [23]. Let n ∈ ℕ and let X be a linear space over the field 𝕂, where 𝕂 is the field of real or complex numbers of dimension d, where d ≥ n ≥ 2. A real valued function ∥·, …, ·∥ on Xⁿ satisfying the following four conditions

(1)
∥x₁, x₂, …, x_n∥ = 0 if and only if x₁, x₂, …, x_n are linearly dependent in X;
(2)
∥x₁, x₂, …, x_n∥ is invariant under permutation;
(3)
∥αx₁, x₂, …, x_n∥ = |α | ∥x₁, x₂, …, x_n∥ for any α ∈ 𝕂;
(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 over the field 𝕂.

For example, if we may take X = ℝⁿ being equipped with the n-norm

= the volume of the n-dimensional parallelepiped spanned by the vectors x₁, x₂, …, x_n which may be given explicitly by the formula

()

where x_i = (x_i1, x_i2, …, x_in) ∈ ℝⁿ for each i = 1,2, …, n, leting (X, ∥·, …, ·∥) be an n-normed space of dimension d ≥ n ≥ 2 and {a₁, a₂, …, a_n} be linearly independent set in X, then the following function ∥·,…,·∥_∞ on Xⁿ⁻¹ defined by

()

defines an (n − 1)-norm on X with respect to {a₁, a₂, …, a_n}.

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

()

A sequence (x_k) in an n-normed space (X, ∥·, …, ·∥) is said to be Cauchy 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.

Let ℳ = (M_k) be a Musielak-Orlicz function, and let p = (p_k) be a bounded sequence of positive real numbers. We define the following sequence spaces in the present paper:

()

If we take ℳ(x) = x, we get

()

If we take p = (p_k) = 1 for all k ∈ ℕ, we have

()

The following inequality will be used throughout the paper. If 0 ≤ p_k ≤ sup p_k = H, K = max (1, 2^H−1), then

()

for all k and a_k, b_k ∈ ℂ. Also

for all a ∈ ℂ.

In this paper, we introduce sequence spaces defined by a Musielak-Orlicz function over n-normed spaces. We study some topological properties and prove some inclusion relations between these spaces.

2. Main Results

Theorem 1. Let ℳ = (M_k) be a Musielak-Orlicz function, and let p = (p_k) be a bounded sequence of positive real numbers, then the spaces , and are linear spaces over the field of complex number ℂ.

Proof. Let , and let α, β ∈ ℂ. In order to prove the result, we need to find some ρ₃ such that

()

Since , there exist positive numbers ρ₁, ρ₂ > 0 such that

()

Define ρ₃ = max (2 | α | ρ₁, 2 | β | ρ₂). Since (M_k) is nondecreasing, convex function and by using inequality (20), we have

()

Thus, we have

. Hence,

is a linear space. Similarly, we can prove that w^θ(ℳ, Λ, p, s, ∥·, …, ·∥) and

are linear spaces.

Theorem 2. Let ℳ = (M_k) be a Musielak-Orlicz function, and let p = (p_k) be a bounded sequence of positive real numbers. Then is a topological linear space paranormed by

()

where H = max (1, sup _kp_k) < ∞.

Proof. Clearly g(x) ≥ 0 for . Since M_k(0) = 0, we get g(0) = 0. Again if g(x) = 0, then

()

This implies that for a given ϵ > 0, there exist some ρ_ϵ(0 < ρ_ϵ < ϵ) such that

()

Thus,

()

Suppose that (x_k) ≠ 0 for each k ∈ ℕ. This implies that Λ_k(x) ≠ 0 for each k ∈ ℕ. Let ϵ → 0, then

()

It follows that

()

which is a contradiction. Therefore, Λ_k(x) = 0 for each k, and thus (x_k) = 0 for each k ∈ ℕ. Let ρ₁ > 0 and ρ₂ > 0 be the case such that

()

Let ρ = ρ₁ + ρ₂; then, by using Minkowski′s inequality, we have

()

Since ρ, ρ₁, and ρ₂ are nonnegative, we have

()

Therefore, g(x + y) ≤ g(x) + g(y). Finally we prove that the scalar multiplication is continuous. Let μ be any complex number. By definition,

()

Thus,

()

where 1/t = ρ/|μ|. Since

, we have

()

So the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem.

Theorem 3. Let ℳ = (M_k) be a Musielak-Orlicz function. If for all fixed x > 0, then .

Proof. Let ; then there exists positive number ρ₁ such that

()

Define ρ = 2ρ₁. Since (M_k) is nondecreasing and convex and by using inequality (20), we have

()

Hence,

Theorem 4. Let 0 < inf p_k = h ≤ p_k ≤ sup p_k = H < ∞ and let be Musielak-Orlicz functions satisfying Δ₂-condition, then one has

(i)
;
(ii)
w^θ(ℳ′, Λ, p, s, ∥·, …, ·∥) ⊂ w^θ(ℳ∘ℳ′, Λ, p, s, ∥·, …, ·∥);
(iii)
.

Proof. Let , then we have

()

Let ϵ > 0 and choose δ with 0 < δ < 1 such that M_k(t) < ϵ for 0 ≤ t ≤ δ. Let for all k ∈ ℕ. We can write

()

So, we have

()

For y_k > δ, y_k < y_k/δ < 1 + y_k/δ. Since

are nondecreasing and convex, it follows that

()

Since ℳ = (M_k) satisfies Δ₂-condition, we can write

()

Hence,

()

From (40) and (43), we have

. This completes the proof of (i). Similarly we can prove that

()

Theorem 5. Let 0 < h = inf p_k = p_k < sup p_k = H < ∞. Then for a Musielak-Orlicz function ℳ = (M_k) which satisfies Δ₂-condition, one has

(i)
;
(ii)
w^θ(Λ, p, s, ∥·, …, ·∥) ⊂ w^θ(ℳ, Λ, p, s, ∥·, …, ·∥);
(iii)
.

Proof. It is easy to prove, so we omit the details.

Theorem 6. Let ℳ = (M_k) be a Musielak-Orlicz function and let 0 < h = inf p_k. Then if and only if

()

for some t > 0.

Proof. Let . Suppose that (45) does not hold. Therefore, there are subinterval I_r(j) of the set of interval I_r and a number t₀ > 0, where

()

such that

()

Let us define x = (x_k) as follows:

()

Thus, by (47),

. But

. Hence, (45) must hold.

Conversely, suppose that (45) holds and let . Then for each r,

()

Suppose that

. Then for some number ϵ > 0, there is a number k₀ such that for a subinterval I_r(j), of the set of interval I_r,

()

From properties of sequence of Orlicz functions, we obtain

()

which contradicts (45), by using (49). Hence, we get

()

This completes the proof.

Theorem 7. Let ℳ = (M_k) be a Musielak-Orlicz function. Then the following statements are equivalent:

(i)
;
(ii)
;
(iii)
.

Proof. (i) ⇒ (ii). Let (i) hold. To verify (ii), it is enough to prove

()

Let

. Then for ϵ > 0, there exists r ≥ 0, such that

()

Hence, there exists K > 0 such that

()

So, we get

(ii) ⇒ (iii). Let (ii) hold. Suppose (iii) does not hold. Then for some t > 0

()

and therefore we can find a subinterval I_r(j), of the set of interval I_r, such that

()

Let us define x = (x_k) as follows:

()

Then . But by (57), , which contradicts (ii). Hence, (iii) must holds.

(iii) ⇒ (i). Let (iii) hold and suppose that . Suppose that ; then

()

Let t = ∥Λ_k (x)/ρ, z₁, z₂, …, z_n−1∥ for each k; then by (59),

()

which contradicts (iii). Hence, (i) must hold.

Theorem 8. Let ℳ = (M_k) be a Musielak-Orlicz function. Then the following statements are equivalent:

(i)
;
(ii)
;
(iii)
.

Proof. (i) ⇒ (ii). It is obvious.

(ii) ⇒ (iii). Let (ii) hold. Suppose that (iii) does not hold. Then

()

and we can find a subinterval I_r(j), of the set of interval I_r, such that

()

Let us define x = (x_k) as follows:

()

Thus, by (62), , but , which contradicts (ii). Hence, (iii) must hold.

(iii) ⇒ (i). Let (iii) hold. Suppose that . Then

()

Again suppose that

; for some number ϵ > 0 and a subinterval I_r(j), of the set of interval I_r, we have

()

Then from properties of the Orlicz function, we can write

()

Consequently, by (64), we have

()

which contradicts (iii). Hence, (i) must hold.

Theorem 9. Let 0 ≤ p_k ≤ q_k for all k and let (q_k/p_k) be bounded. Then

()

Proof. Let x = (x_k) ∈ w^θ(ℳ, Λ, q, s, ∥·, …, ·∥); write

()

and μ_k = p_k/q_k for all k ∈ ℕ. Then 0 < μ_k ≤ 1 for all k ∈ ℕ. Take 0 < μ ≤ μ_k for k ∈ ℕ. Define sequences (u_k) and (v_k) as follows.

For t_k ≥ 1, let u_k = t_k and v_k = 0, and for t_k < 1, let u_k = 0 and v_k = t_k. Then clearly for all k ∈ ℕ, we have

()

Now it follows that

and

. Therefore,

()

Now for each k,

()

and so

()

Hence, x = (x_k) ∈ w^θ(ℳ, Λ, p, s, ∥·, …, ·∥). This completes the proof of the theorem.

Theorem 10. ( i) If 0 < inf p_k ≤ p_k ≤ 1 for all k ∈ ℕ, then

()

( ii) If 1 ≤ p_k ≤ sup p_k = H < ∞, for all k ∈ ℕ, then

()

Proof. (i) Let x = (x_k) ∈ w^θ(ℳ, Λ, p, s, ∥·, …, ·∥); then

()

Since 0 < inf p_k ≤ p_k ≤ 1, this implies that

()

therefore,

()

Hence,

()

(ii) Let p_k ≥ 1 for each k and sup p_k < ∞. Let x = (x_k) ∈ w^θ(ℳ, Λ, s, ∥·, …, ·∥); then for each ρ > 0, we have

()

Since 1 ≤ p_k ≤ sup p_k < ∞, we have

()

Therefore, x = (x_k) ∈ w^θ(ℳ, Λ, p, s, ∥·, …, ·∥), for each ρ > 0. Hence,

()

This completes the proof of the theorem.

Theorem 11. If 0 < inf p_k ≤ p_k ≤ sup p_k = H < ∞, for all k ∈ ℕ, then

()

Proof. It is easy to prove so we omit the details.

Acknowledgment

The authors are very grateful to the referees for their valuable suggestions and comments. The third author also acknowledges the partial support by University Putra Malaysia under the project ERGS 1-2013/5527179.

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Sequence Spaces Defined by Musielak-Orlicz Function over n-Normed Spaces

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Sequence Spaces Defined by Musielak-Orlicz Function over n-Normed Spaces

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