Continuous Linear Operators On Infinite Quasi-Sobolev Spaces ℓ∞m

In this study, the concept of infinite quasi-Sobolev spaces ℓ∞m , where m ∈ ℝ is considered. These spaces have been proved as quasi-Banach spaces, as well as, Banach spaces, while they neither Hilbert spaces nor quasi-Hilbert spaces. Some kinds of linear operators such as continuous, bounded, closed and completely continuous for operators which map ℓ∞m or ℓ1m into ℓ∞m are discussed.


Introduction and Preliminaries
Quasi-normed space   || . || , q U or simply U is a real vector space with a quasi-norm || . || q , which is a positive definite, absolutely homogeneous functional such that there is a constant K ∈ [1, ∞ ) , Also, a function || . || q be a norm || . || if K= 1, thus it is a generalization of a norm function.
Definitely, a quasi-normed space U is metrizable, thus the concept of completeness is correct, and it is called a quasi-Banach space [1,2].
Related to a quasi-normed space U, is an inner product space if and only if the following equation is hold: If the equality: is satisfied, then U is said to be a quasi-inner product, where (v, w) and (w, v) are Gateaux derivatives, A Ĝteaux derivative of where h ∊ ℝ . Similarly, (w, v) at w ∈ U in the direction v is defined. If U is a quasi-Banach space, then it is called a quasi-Hilbert space [3].
In [4], a sequence {2 kσ }, ∈ ℕ used to define a space ℓ p σ (A), where A is a set of sequences, 1≤ p ≤ ∞, σ is a real number.
In [5,6], we have been used a set of all monotonically increasing eigen values {λ k } ⊂ℝ + such that lim →∞ λ k = + ∞, of an operator which was defined on Sobolev spaces to construct quasi-Sobolev spaces ℓ p , where 0 < p < ∞ and m ∈ ℝ which are defined as : Also, a sequence {λ k } was used to define some types of operators on these spaces.
In this study, we devote transference above ideology using {λ k } to construct sequence space ℓ ∞ and to define continuous operators on these spaces. For every m ∈ ℝ, ℓ ∞ is called an infinite quasi-Sobolev space and is defined as: Theorem 1.1 [5,6]. Sequence spaces ℓ p , 0 < p < ∞, are quasi-Banach spaces, and they are Banach spaces only when 1 ≤ p < ∞.
Remark 1.2 [3]. Not all spaces ℓ p , where 0 < p < ∞, are quasi-Hilbert spaces, such as, ℓ 1/2 , ℓ 3 , while ℓ 4 be a quasi-Hilbert space, where a functional (v, w) in ℓ p defines as:   In the second section of this work, a proof of an infinite quasi-Sobolev spaces ℓ ∞ , for every m ∈ ℝ as quasi-Banach space is confirmed, while it is not a quasi-Hilbert space and a relationship between ℓ 1 and ℓ ∞ is presented, while in the third section, a linear operator which is defined on ℓ ∞ or ℓ 1 , is proved as continuous, imply it is closed. Also, continuity of an operator is insufficient to be completely continuous operator.

An Infinite Quasi-Sobolev Spaces
In this part, we review Banach, Hilbert, quasi-Banach and quasi-Hilbert space for a sequence space ℓ ∞ , with given the relationship between ℓ 1 and ℓ ∞ .

Proof.
A positive definite property and an absolute homogeneous property are obvious. Since,  (2) it is an inner-product space if and only if the following equivalence holds:

Continuous Linear Operators
In this section, we use equivalence of boundedness and continuity for linear operators on Banach spaces. Let {λ k } ⊂ℝ + is a monotonically increasing sequence such that lim →∞ λ k = + ∞. Theorem 3.1. An operator : ℓ ∞ → ℓ ∞ , m ∈ ℝ such that Tu = λ k −1 u k , ∈ ℕ is a continuous linear operator.
, then that T is bounded, and it is continuous.
Remark 3.5. Every completely continuous operator is continuous, since discontinuity of an operator T which is defined on ℓ 1 or ℓ ∞ into ℓ ∞ would imply existence a sequence { k } such that ∥ k ∥ ≤1 and ∥ k ∥→ ∞ and this implies that T is not completely continuous . Conversely, may be not true, as shown in the following example: Example 3.6. Consider a linear operator T: ℓ ∞ → ℓ ∞ , Tu = u . Suppose { k } such that k = 2 k is any bounded sequence in ℓ ∞ where { k } is a sequence of all zeroes except in the k-th spot where there appears 1 and ∥ k ∥ = 1, then, for any k ≠ r, ∥ T k − T r ∥=2∥ k − r ∥= 2 . Hence, T is continuous, and also it is closed. But this operator is not completely continuous, because, any subsequence of { k } is not converge.

Acknowledgements
The author would like to express thanks to College of Science-Musyansiriyah University for supporting this work. Special thanks to unknown referees for their careful reading and helpful comments.