On Some Geometric Properties of a New Paranormed Sequence Space

In literature, there are many papers about geometric properties and their applications on different sequence spaces. Some of them are as follows. In [1], Opial defined the Opial property with his name mentioned and he proved that l p (1 < p < ∞) satisfies this property but the space L p [0, 2π] (p ̸ = 2, 1 < p < ∞) does not. Franchetti [2] has shown that any infinite dimensional Banach space has an equivalent norm satisfying the Opial property. Later, Prus [3] has introduced and investigated uniform Opial property for Banach spaces. In [4], the notion of nearly uniform convexity for Banach spaces was introduced by Huff. Also Huff proved that every nearly uniformly convex space is reflexive and it has uniform Kadec-Klee property. However, Kutzarova [5] defined knearly uniformly convex Banach spaces. Shue [6] first defined Cesaro sequence spaces with norm. In [7], it is shown that the Cesaro sequence spaces ces p (1 ≤


Introduction
In literature, there are many papers about geometric properties and their applications on different sequence spaces.Some of them are as follows.
Franchetti [2] has shown that any infinite dimensional Banach space has an equivalent norm satisfying the Opial property.Later, Prus [3] has introduced and investigated uniform Opial property for Banach spaces.
In [4], the notion of nearly uniform convexity for Banach spaces was introduced by Huff.Also Huff proved that every nearly uniformly convex space is reflexive and it has uniform Kadec-Klee property.However, Kutzarova [5] defined nearly uniformly convex Banach spaces.
In [8], it was shown that Banach-Saks of type- property holds in these spaces.
Later, Sanhan and Suantai [9] generalized the normed sequence spaces to the paranormed sequence spaces.He showed that the sequence spaces ces() equipped Luxemburg norm are rotund and have Kadec-Klee property.
Petrot and Suantai [10] studied the uniform Opial property of these spaces.In [9], Sanhan and Suantai have showed that the Cesaro sequence space ces(), where the sum runs over 2  ≤  ≤ 2 +1 , equipped with Luxemburg norm has property () but it is not rotund.
Karakaya [11] introduced a new sequence space involving lacunary sequences connected with Cesaro sequence space and examined some geometric properties of this space equipped with Luxemburg norm.In [12], Karakas ¸et al. defined and studied a new difference sequence space involving lacunary sequences by using difference operator.
In [15], S ¸ims ¸ek and Karakaya generalized sequence space ces[(  ), (  )] to vector-valued space ces(,   ,   ) and investigated some topological and geometrical properties as Kadec-Klee and rotund according to Luxemburg norm of this space.
In [16], Savas ¸et al. introduced an ℓ  -type new sequence space and examined some geometrical properties of this

Preliminaries and Notation
Let (, ‖ ⋅ ‖) (for the brevity  = (, ‖ ⋅ ‖)) be a normed linear space and let () (resp.()) be the closed unit ball (resp.unit sphere) of .The space of all real sequences is denoted by .For any sequence {  } in , we denote by conv({  }) the convex hull of the elements of {  }.
A Banach space  is called uniformly convex (UC) if for each  > 0, there is  > 0 such that, for ,  ∈ (), the inequality ‖ − ‖ >  implies that Recall that for a number  > 0 a sequence {  } is said to be an A Banach space  is said to have the Kadec-Klee property (H property) if every weakly convergent sequence on the unit sphere is convergent in norm.
A Banach space  is said to have the uniform Kadec-Klee property (UKK) if for every  > 0 there exists  > 0 such that if  is the weak limit of a normalized -separated sequence, then ‖‖ < 1 −  (see [4]).We have that every (UKK) Banach space has the Kadec-Klee property.
A Banach space  is said to be the nearly uniformly convex (NUC) if for every  > 0 there exists  > 0 such that, for every sequence {  } ⊂ () with sep({  }) > , we have Let  ≥ 2 be an integer.
Of course a Banach space  is (NUC) whenever it is (-NUC) for some integer  ≥ 2. Clearly, (-NUC) Banach spaces are (NUC) but the opposite implication does not hold in general (see [5]).
A Banach space  is said to have the Opial property if every sequence {  } that is weakly convergent to  0 satisfies lim for every  ∈  and  ̸ =  0 (see [1]).
A Banach space  is said to have the uniform Opial property if every  > 0 there exists  > 0 such that, for each weakly null sequence {  } ⊂ () and  ∈  with ‖‖ ≥ , we have (see [3]) A point  ∈ () is called an extreme point if for any ,  ∈ () the equality 2 =  +  implies that  = .A Banach space  is said to be rotund (abbreviated as ()) if every point of () is an extreme point.A Banach space  is said to be  -rotund (written as kR) (see [19]) if for every sequence implies that {  } is convergent.
It is well known that (UC) implies (kR) and (kR) implies (( + 1)), and (kR) spaces are reflexive and rotund, and it is easy to see that (-NUC) implies (kR).
For any modular  on , the space is called the modular space.
A sequence (  ) of elements of   is called modular convergent to  ∈   if there exists a  > 0 such that ((  − )) → 0 as  → ∞.If  is a convex modular, then the following formula defines a norm on   which is called the Luxemburg norm: A modular  is said to satisfy the Δ 2 -condition ( ∈ Δ 2 ) if for any  > 0 there exist constants  ≥ 2 and  > 0 such that for all  ∈   with () ≤ .
See [20].In this paper, we will need the following inequalities in the sequel: for  ≥ 1.

Main Results
In this section, we will give some basic properties of the modular  on the space ℓ Δ (, V, ).Also, we will investigate some relationships between the modular  and the Luxemburg norm on ℓ Δ (, V, ).Finally, we study some geometric properties on this space.Let us start with some lemmas which will be used in the proof of the theorems about geometric properties of this space.Lemma 6.The functional  is a convex modular on ℓ Δ (, V, ).
Lemma 7.For any  ∈ ℓ Δ (, V, ), Proof.It can be proved with standard techniques in a similar way as in [23].
Lemma 8. Let {  } be a sequence in ℓ Δ (, V, ): Proof.It can be proved with standard techniques in a similar way as in [23].
Proof.Since  = (  ) is bounded, it is easy to see that  ∈ Δ  2 .Hence, the lemma is obtained directly from Lemma 1.
Proof.Since  ∈ Δ  2 , the lemma is obtained directly from Lemma 2. Now we will show that the ℓ Δ (, V, ) is a Banach space with respect to the Luxemburg norm Theorem 12.The space ℓ Δ (, V, ) is a Banach space with respect to the Luxemburg norm defined by Proof.We will show that every Cauchy sequence in ℓ Δ (, V, ) is convergent according to the Luxemburg norm.Let (   ) be a Cauchy sequence ℓ Δ (, V, ) and  ∈ (0, 1).Thus, there exists  0 () such that ‖  −   ‖ <  for all ,  ≥  0 .By the Lemma 8(i), we obtain for all ,  ≥  0 (); that is, For fixed  we get that Hence, we obtain that the sequence (  ()) is a Cauchy sequence in R. Since the real number R is complete,   () → () as  → ∞.Therefore, for fixed  and So, we obtain that for all  ≥ n 0 () and as  goes to infinity So, for all  ≥  0 () from Lemma 8(i), It can be seen that, for all  ≥  0 ,   →  and (  − ) ∈ ℓ Δ (, V, ).
From the linearity of the sequence space ℓ Δ (, V, ), we can write that Hence, the sequence space ℓ Δ (, V, ) is a Banach space with respect to the Luxemburg norm.This completes the proof of the theorem.
Now, we shall give the main theorems of this paper involving the geometric properties of the space ℓ Δ (, V, ).Theorem 14.The space ℓ Δ (, V, ) has the Kadec-Klee property.