Some Topological and Geometric Properties of Some New Spaces of λ-Convergent and Bounded Series

The main purpose of this study is to introduce the spaces cs, cs 0 , and bs which are BK-spaces of nonabsolute type. We prove that these spaces are linearly isomorphic to the spaces cs, cs 0 , and bs, respectively, and derive some inclusion relations. Additionally, Schauder bases of the spaces cs and cs 0 have been constructed and the α-, β-, and γ-duals of these spaces have been computed. Besides, we characterize some matrix classes from the spaces cs, cs 0 , and bs to the spaces lp, c, and c0, where 1 ≤ p ≤ ∞. Finally, we examine some geometric properties of these spaces as Gurařı’s modulus of convexity, property m∞, property (M), property WORTH, nonstrict Opial property, and weak fixed point property.


Introduction
By a sequence space, we understand a linear subspace of the space  = C N , where N = {0, 1, 2, . ..} and C denotes the complex field.A sequence space  with a linear topology is called a -space provided each of the maps   :  → C defined   () =   which is continuous for all  ∈ N. A space is called an -space provided  is a complete linear metric space.An -space whose topology is normable is called a -space (see [1, pages 272-273]) which contains , the set of all finitely nonzero sequences.We write ℓ ∞ , , and  0 for the spaces of all bounded, convergent, and null sequences, respectively.Also by ℓ  , we denote the space of all absolutely summable sequences, where 1 ≤  < ∞.Moreover, we write , , and  0 for the spaces of all bounded, convergent, and null series, respectively.
Let  and ] be two sequence spaces, and let  = (  ) be an infinite matrix of complex numbers   , where ,  ∈ N. Then we say that  defines a matrix transformation from  into ], and we denote it by writing  :  → ] if for every sequence  = (  ) ∈ , the sequence  = {()  }, the transform of , is in ], where ()  := ∑      , ( ∈ N,  ∈  00 ()) , (1) and by  00 () denotes the subspace of  consisting of  ∈  for which the sum exists as a finite sum.For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞ and we will use the convention that any term with a negative subscript is equal to naught; for example,  −1 = 0 and  −1 = 0.By ( : ]), we denote the class of all matrices  such that  :  → ].Thus  ∈ ( : ]) if and only if the series on the right side of (1) converges for each  ∈ N and each  ∈  and we have  = {()  } ∈N ∈ ] for all  ∈ .For an arbitrary sequence space , the matrix domain   of an infinite matrix  in  is defined by which is a sequence space.If  is triangle, then one can easily observe that the normed sequence spaces   and  are norm isomorphic; that is,   ≅ .If  is a sequence space, then the continuous dual  *  of the space   is defined by  *  := { :  =  ∘ ,  ∈  * } .
We denote the collection of all finite subsets of N by F. Also, we will write  () for the sequence whose only nonzero term is 1 in the th place for each  ∈ N. Throughout this paper, Journal of Function Spaces let  = (  ) be a strictly increasing sequence of positive real numbers tending to infinity; that is, We define the matrix Λ = (  ) of weighted mean relative to the sequence  by for all ;  ∈ N.With a direct calculation we derive the equality It is easy to show that the matrix Λ is regular and is reduced, in the special case   =  + 1 for all  ∈ N to the matrix  1 of Cesàro means of order one.Introducing the concept of Λstrong convergence, several results on Λ-strong convergence of numerical sequences and Fourier series were given by Móricz [2].Since we have in the special case =   − −1 for all  ∈ N, the matrix Λ is also reduced to the Riesz means   = (  ) with respect to the sequence  = (  ).
We summarize the knowledge in the existing literature concerning domain of the matrix  over some sequence spaces.Mursaleen and Noman [3][4][5][6] introduced the spaces ℓ  ∞ ,   ,   0 , and ℓ   of lambda-bounded, lambda-convergent, lambda-null, and lambda-absolutely -summable sequences and gave the inclusion relations between these spaces and the classical sequence spaces ℓ ∞ , , and  0 .Later, Mursaleen and Noman [7] investigated the difference spaces   0 (Δ) and   (Δ) obtained from the spaces   0 and   .Recently, paranormed -sequence spaces of nonabsolute type have been studied by Karakaya et al. [8].More recently, Sönmez and Bas ¸ar [9] introduce the difference sequence spaces   0 () and   (), which are the generalization of the spaces   0 (Δ) and   (Δ).Quite recently, some new sequence spaces of nonabsolute type and matrix transformations have been studied by Ganie and Sheikh [10].The same authors have studied the spaces of -convergent sequences and almost convergence [11].Also, the fine spectrum of the operator defined by lambda matrix over the spaces of null and convergent sequences has been studied by Yes ¸ilkayagil and Bas ¸ar [12].
In this work, our purpose is to construct the sequence spaces   0 ,   , and   by the domain of the matrix Λ in the spaces  0 , , and , respectively, of the series whose sequence of partial sums are in the spaces  0 , , and ℓ ∞ [3].
We define the sequence  = (  ) by the Λ-transform Λ of a sequence  = (  ); that is,  = Λ, and so we have Also, we say that a sequence  = (  ) ∈  is -convergent if Λ ∈ .In particular, we say that  is -null or -bounded if Λ ∈  0 or ℓ ∞ , respectively.
2. The Sequence Spaces   ,   0 , and In the present section, we introduce the sequence spaces   ,   0 , and   as the sets of all sequences whose Λ-transforms are in the spaces ,  0 , and , respectively; that is, With the notation of (2), we can redefine the spaces   ,   0 , and   as the matrix domains of the triangle Λ in the spaces ,  0 , and  by Then, it is immediate by (12) that the sets   ,   0 , and   are linear spaces with coordinatewise addition and scalar multiplication; that is,   ,   0 , and   are the sequence spaces consisting of all sequences which are -convergent, null, and -bounded series of type , respectively.Now, we may begin with the following theorem which is essential in the text.Proof.To prove this, we should show the existence of an isometric isomorphism between the spaces   0 and  0 .Consider the transformation  defined, with the notation of (8), from   0 to  0 by   →  = .Then,  =  = Λ ∈  0 for every  ∈   0 and the linearity of  is clear.Also, it is trivial that  =  whenever  =  and hence  is injective.Furthermore, let  = (  ) ∈  0 be given and define the sequence  = (  ) by Then, by using ( 8) and ( 14), we have for every  ∈ N that This shows that Λ =  and since  ∈  0 , we obtain that Λ ∈  0 .Thus, we deduce that  ∈   0 and  = .Hence  is surjective.Moreover, one can easily see for every  ∈   0 that which means that  is norm preserving.Therefore  is isometry.Consequently  is an isometric isomorphism which shows that the spaces   0 and  0 are isometrically isomorphic.
It is clear that if the spaces   0 and  0 are replaced by the respective one of the spaces   and  or   and , then we obtain the fact that   ≅  and   ≅ .This completes the proof.

The Inclusion Relations
In the present section, we establish some inclusion relations concerning the spaces   ,   0 , and   .We may begin with the following lemma.
Proof.It is obvious that the inclusions   0 ⊂   ⊂   hold.Let us consider the sequence  = (  ) defined by In the present case, we obtain for every  ∈ N that which shows that Λ ∈  \  0 .Thus, the sequence  is in   but not in   0 .Hence   0 ⊂   is a strict inclusion.To show the strictness of the inclusion   ⊂   , we define the sequence  = (  ) by Then, we have for every  ∈ N that This shows Λ ∈ \.Thus, the sequence  is in   but not in   and hence   ⊂   is a strict inclusion.This concludes the proof.
Proof.It is clear that the inclusion   ⊂   0 holds, since  ∈   implies Λ ∈  and hence Λ ∈  0 which means that  ∈   0 .Consider the sequence  = (  ) defined by Then,  ∈  0 and hence  ∈   0 , since the inclusion  0 ⊂   0 holds.On the other hand, we have for every  ∈ N that which shows that Λ ∉  and hence  ∉   .Thus, the sequence  is in   0 but not in   .Therefore, the inclusion   ⊂   0 is strict.Similarly, it is also trivial that the inclusion   ⊂ ℓ  ∞ holds.To show that this inclusion is strict, we define the sequence  = (  ) by  =  = (1, 1, 1, . ..).In the present case, we have for every  ∈ N that which shows that Λ ∈ ℓ ∞ \ .Thus, the sequence  is in ℓ  ∞ but not in   and hence   ⊂ ℓ  ∞ is a strict inclusion.This completes the proof.
Conversely, let  ∈   be given.Then, we have by the hypothesis that  ∈ .Again, it follows by (26) that which shows that  ∈  while Λ ∈  and  ∈ .Hence, the inclusion   ⊂  holds and this concludes the proof.Proof.Suppose that the inclusion   ⊂  holds, and take any  = (  ) ∈   .Then,  ∈  by the hypothesis.Thus, we obtain from equality ( 17) which yields that  ∈ .
Conversely, assume that  ∈  for every  ∈   .Again, we obtain from equality ( 17) This shows that  ∈ .Hence, the inclusion   ⊂  holds.This completes the proof.

4.
The Basis for the Spaces   ,   0 , and In the present section, we give a sequence of the points of the spaces   and   0 which forms a basis for these spaces.If a normed sequence space  contains a sequence (  ) with the property that for every  ∈  there is a unique sequence of scalars (  ) such that lim then (  ) is called a Schauder basis (or briefly basis) for .
The series ∑      which has the sum  is then called the expansion of  with respect to (  ) and is written as  = ∑      .Now, since the transformation  defined from   0 to  0 in the proof of Theorem 3 is an isomorphism, we have the following theorem.

Theorem 11. Define the sequence 𝑒
for all  ∈ N.
Then, one has the following: (a) The sequence ( (0)  ,  (1)   , . ..) is a Schauder basis for the spaces   and   0 and every  ∈   or   0 has a unique representation of the form (b)   has no Schauder basis.
Finally, let us show the uniqueness of the representation (33) of  ∈   .Suppose on the contrary that there exists another representation  = ∑     ()   .Since the linear transformation  defined from   to , in the proof of Theorem 3, is continuous, we have Therefore, the representation (33) of  ∈   is unique.It can be proved similarly for   0 .This completes the proof.(b) As a direct consequence of Remark 2.2. of Malkowsky and Rakocevic [14], since  has no Schauder basis   also has no Schauder basis.
As a result, it easily follows from Theorem 1 that   and   0 are the Banach spaces with their natural norms.Then, by Theorem 11 we obtain the following corollary.In this section, we state and prove the theorems determining the -, -, and -duals of the sequence spaces   ,   0 , and   of nonabsolute type.For arbitrary sequence spaces  and , the set (, ) defined by is called the multiplier space of  and .One can easily observe for a sequence space  with  ⊂  and  ⊂  that the inclusions (, ) ⊂ (, ) and (, ) ⊂ (, ) hold, respectively.With the notation of (39), the -, -, and duals of a sequence space , which are, respectively, denoted by   ,   , and   , are defined by It is clear that   ⊂   ⊂   .Also, it can be obviously seen that the inclusions   ⊂   ,   ⊂   , and   ⊂   hold whenever  ⊂ .
The following known results [15] where the matrix   = (   ) is defined via the sequence  = (  ) ∈  by for all ,  ∈ N. Then {  }  =   1 and Proof.Let  = (  ) ∈ .Then, by bearing in mind relations ( 8) and ( 14), it is immediate that the equality where Proof.Because the proof may also be obtained for the space   in a similar way, we omit it.Take any  = (  ) ∈  and consider the equation where the matrix   = (   ) is defined by )   , (60) respectively.Thereby, we conclude that Theorem 24.The -dual of the spaces   ,   0 , and   is the set   3 ∩   4 .
Proof.The proof of this result follows the same lines as those in the proof of Theorem 23 using Lemmas 19, 20, and 21 instead of Lemma 16.
Lemma 25.Consider  = (  ) ∈ ( :  0 ) if and only if (44) holds and Proof.Suppose that conditions (67) and (68) hold and take any  = (  ) ∈   .Then, we have by Theorem 23 that (  ) ∞ =0 ∈ (  )  for all  ∈ N and this implies the existence of the -transform of ; that is,  exists.Further, it is clear that the associated sequence  = (  ) is in  and hence  ∈  0 .
Let us now consider the following equality derived by using relation (8) from the th partial sum of the series Therefore, by using (67) and (68), from (71) as  → ∞, we obtain that

Some Geometric Properties of the Spaces
,   0 , and In this section, we investigate some geometric properties for the sequence spaces   and   .Let (, ‖ ⋅ ‖) be a normed linear space, and let () and () be the unit sphere and unit ball of  (for the brevity  = (, ‖ ⋅ ‖)), respectively.Consider Clarkson's modulus of convexity (Clarkson [16] and Day [17]) defined by where 0 ≤  ≤ 2. The inequality   > 0 for all  ∈ (0, 2] characterizes the uniformly convex spaces.
In [18], Gurarǐ's modulus of convexity is defined by It remains unknown if property WORTH implies fixed point property.In many situations, the fixed point property can be easily obtained when we assume, in addition, that the spaces are considered to have the property WORTH.
The following result will be used in our results.(iv)  satisfies the nonstrict Opial property.
Now, let us give our first theorem in this section.

Theorem 1 .
The sequence spaces   ,   0 , and   are BKspaces with the same norm ‖‖   = ‖‖   0 = ‖‖   ; that is,‖‖   = ‖Λ‖  = supProof.Since (12) holds and ,  0 , and  are -spaces with the respect to their natural norms and the matrix Λ is a triangle, Theorem 4.3.12 of Wilansky[13, page 63]gives the fact that   ,   0 , and   are -spaces with the given norms.This completes the proof.The sequence spaces   ,   0 , and   of nonabsolute type are isometrically isomorphic to the spaces ,  0 , and , respectively; that is,   ≅ ,   0 ≅  0 , and   ≅ .

Theorem 9 .
The inclusion   0 ⊂  0 holds if and only if  ∈  0 for every sequence  ∈   0 .Proof.One can see by analogy to Theorem 8 that the inclusion   0 ⊂  0 also holds if and only if  ∈  0 for every sequence  ∈   0 .This completes the proof.The inclusion   ⊂  holds if and only if  ∈  for every sequence  ∈   .
are fundamental for this section.
then for each  ∈  we have that limsup  → ∞ ‖  − ‖ is an increasing function of  on [0, ∞).