Generalized Köthe-Toeplitz Duals of Some Vector-Valued Sequence Spaces

We know from the classical sequence spaces theory that there is a useful relationship between continuous and ββ-duals of a scalarvalued FK-space EE originated by the AK-property. Our main interest in this work is to expose relationships between the operator space L(EEEEEE and EE and the generalized ββ-duals of some XX-valued AK-space EE where XX and EE are Banach spaces and EE = {(AAkkEE AAkk ∈ L(XXEEEE X X ∞ kk=k AAkkxxkk converges in EEE for all xx ∈ EEx. Further, by these results, we obtain the generalized ββ-duals of some vector-valued Orlicz sequence spaces.

to show that there is an analogue relationship for -valued sequence spaces in the context of generalized -duals with respect to another �xed Banach space .Further, by applying this result, we obtain generalized -duals of some vectorvalued Orlicz sequence spaces.We think that our results give a fruitful way to �nd generalized duals of this kind of vectorvalued sequence spaces.

Prerequisites
We use the notations ℕ, ℂ, and ℝ for the sets of all positive integers, complex numbers, and real numbers, respectively.For some locally convex (lc, for short) space ,  * denotes the continuous dual of  and we denote by   and   the closed unit ball and the sphere of some normed space , respectively.
An FH-space is an lc Fréchet space  such that  is a vector subspace of a Hausdorff topological vector space  and the topology of  is larger than the restricted topology of  to ; that is, the inclusion map:    is continuous.If    then an FH-space is called an FK-space.With a little extension, an -valued sequence space  is called an FK-space whenever    where  is a Banach space.In fact, the theory of FK-spaces can be developed without the local convexity.However, we are interested only in locally convex FK-spaces.Note that,    ℕ and so its topology is the weakest topology such that the projections   ∶   ⟶ ,       ,   , ,  (5) are continuous.
An Orlicz function is a function  ∶ ,   ,  which is continuous, nondecreasing, and convex with   ,    for all    and    as   .An Orlicz function  can always be represented in the following integral form: where , known as the kernel of , is right differentiable for   ,   ,    for   , and  is nondecreasing and    as   .
Consider the kernel  associated with Orlicz function , and let    sup  ∶   ⩽  .(7) en  possesses the same properties as the function .
International Journal of Analysis 3

Relative Weak Topologies
An operator    , for Banach spaces  and , is called a Hahn-Banach operator if for every Banach space  containing  as a subspace there exists an operator      such that ‖  ‖  ‖‖ and    , for every   .us, the classical Hahn-Banach theorem can be restated in the following way for operators (see also [5]).eorem 1.Let ,  be Banach spaces and let      be a continuous linear operator of rank 1. en  is a Hahn-Banach operator.
By some modi�cation on the assertion� norm preserving, the result remains true if  is taken as a locally convex space.e evidence of this assertion can be found in [6, Section 7.2.].
Hence, by using eorem 1 we derived some tools for later sections as in the way that is similar to classical treatments.e proof of the following result is also given in [7].Nevertheless, it will be convenient to restate it here.Proof.Let    be   -bounded.We are going to show that  is bounded for each seminorm  in  where  is the family of all seminorms generating the lc topology of .For an arbitrary      is a seminormed space.us we can show as in eorem 6 that   is bounded in ℒ 2  , whence,  is bounded in , that is,  is a bounded subset of ℝ.
We conclude this section with a brief discussion of equicontinuity.A set Ω of linear maps from one topological vector space  into another one  is called equicontinuous if, for each neighborhood  of 0 in , is a neighborhood of 0 in .Equicontinuity is a generalization of the uniform boundedness of the family of linear maps between seminormed spaces.eorem 8 (see [9]).Let Ω be a collection of continuous linear mappings  from the Fréchet space  into the topological vector space .en Ω is equicontinuous if and only if the set is bounded in , for each   .
Lemma 9 (see [6]).Let    be a net of continuous operators   from a Fréchet space  into the topological vector space .en the set { ∶     0} is a closed subspace of .

Sectional Properties and Operator Spaces
For an   , is called th section of .Further,  denotes the space of all �nite sequences in .
De�nition ��.Let    be an FK-space.If, for each   , then  is called an AK-space.Further,  is called an AD-space whenever  is dense in .If, for each   , the sequence    is bounded in  then  is called an AB-space.
Let    be an FK-space and de�ne the set Clearly,    and    whenever  is an AB-space.

Applications on Vector-Valued Orlicz Sequence Spaces
It is not hard to see as in the classical case, [4] Proofs of this lemma and the above assertion can be given in a similar way followed in [4, eorem 8.9], by using the inequality and by using the fact that  =     ℓ   if and only if    ∞ =  ℓ  .
In general ℎ   has no Schauder basis in classical manner.In [10] we introduce a new kind basis notion.Let us give this de�nition and prove that ℎ   has a basis in this manner.
International Journal of Analysis �e�nition 1� (see [10]).Let  and  be Banach spaces and  be a set.A family {  ∶    of continuous linear functions   ∶    is called -basis for  if the following condition is satis�ed.ere exists a directed subset  (by some relation ≪) of ℱ satisfying the property; for each    there is some    such that   , and there exists a unique family {  ∶    of linear functions   from  onto  such that, for each   , the net (  ( [4]Orlicz function  is said to satisfy the Δ  -condition for small  at  if for each    there exist     and     such that  ⩽   , for all   ,   ][4].e space ℓ  consists of all sequences    of scalars such that  is closely related to the space ℓ  which is an Orlicz sequence space with     ,  ⩽  < .Another de�nition of ℓ  ,[4], is given by the complementary function to  as follows:ℓ          ∶   converges, for all    ℓ   , (12)where  is the complementary function to  and  ℓ  is the collection of all  in  with ∑      < .Clearly,  ℓ  ⊆ ℓ  and ℓ  are normed by the Orlicz norm en  is an Orlicz function.efunctionsand  are called mutually complementary Orlicz functions, and they satisfy the Young inequality,  ⩽   +   for ,   .(9)

Corollary 2 .
Let   be Banach spaces and   .en, for some     , there exists a corresponding operator      such that Proof.If  is not null, take   .en  is a closed subspace of  hence is a Banach space with the same norm.�e�ne  ′    ‖‖, for some     , from  into .

Corollary 3. Let 𝑋𝑋 be an lc space, 𝑇𝑇 be a Banach space
,  be a vector subspace of , and     .enthereexistsan     and a corresponding operator    such that          on  (19) Proof.Let    and consider     .Fix some     and de�ne the operator Let   be a topology on  such that, for each net      in        if and only if ‖  ‖    for each    .It is an lc topology generated by the family    ‖ ⋅ ‖  ∘       of the seminorms ‖ ⋅ ‖  ∘  on .Obviously, the norm topology of  is stronger than   , in general.If  is a scalar �eld of  then   coincide with the usual weak topology.It is clear that, a net which is   -convergent to  is also weak convergent to .e converse of this assertion is not true.Example 4. Let      2 .en the sequence    ∞ 1 of unit vectors is weak convergent to  in  2 [8, page 99].But, it is not   2 -convergent to .erefore, for the identity operator on  2 , we have ‖  ‖  ‖  ‖  1 ↛ .However, we cannot work this example in  1 (in fact, in a Banach space which has the Schur property) since weak convergence implies the norm convergence in this case.Hence the following result is obvious from the de�nition of the Schur property.eorem 5. Let  be a Banach space having the Schur property.enweak convergence implies   -convergence in  for every Banach space .Now consider the canonical embedding    2  , where  2   is the space of all continuous operators from   into  and  and  are Banach spaces, which assigns each    to the operator   on   de�ned by      for each         as is in the classical case.Now, let us investigate how do the bounded subsets of the  in the   -topology behave.Note that a subset  of  is called   -bounded if  is bounded in  for each    .It is clear that, for every pair of the Banach spaces  and     is   -bounded if it is norm bounded.econverse of this assertion is the following theorem.Let  and  be Banach spaces.en-bounded sets are norm bounded.Proof.Let    be   -bounded and   be canonical embedding of  into  2  .A hypothesis says that  is pointwise bounded so it is uniformly (norm) bounded by the uniform boundedness principle.Hence there exists a    such that ‖  ‖ ≤  for each   .So Let  be an lc space and  be a Banach space.en   -bounded sets are also bounded in the lc topology of .
[6,    (equivalently, for some    and    ( ℂ or ℝ) such that     ).Clearly, the hyphothesis      says that  is not dense in .us,mustbeclosed in  since it is a maximal subspace of  (see[6, Prob.425]).on .us, the extension   of  ′  is the desired operator.Let us establish an lc topology on a Banach space  with respect to another Banach space .
, for some Banach space , by       ∶            ∀  ℒ   .For each Banach space     .Proof.Let     and   ℒ  then we can write   ∑      , that is, Let    be an FK-space.en  is an AKspace if and only if   and ℒ  are isomorphic for every Banach space .Proof.Let       where each    ℒ  and de�ne   ] =  by       .is means that  is surjective.Conversely, let   and ℒ  be isomorphic.en each   ℒ  has the representation    such that each ; that is, the sequence {  } is   -convergent hence it is   -bounded.us, it is also bounded in the lc Fréchet topology of  by eorem 7. Proposition 12.For each Banach space ,      (34) where  is the closure of  in .Proof.Let     .en, for every   ℒ  such that   0 on ,               0 …  0 en the sequence {  } is bounded by the AB-property.erefore {  } is equicontinuous by eorem 8. On the other hand     0 for each   , at is,          ⟶ 0  Λ.where   ∶     =    is the th (continuous) projection de�ned by    =   .en each   is continuous and the sequence    is pointwise convergent since the series ∑     is convergent.So, the operator , which is also de�ned by  =      (42) is continuous by the Banach-Steinhauss closure theorem, whence,   ℒ .at  is injective comes from the following discussion.Let   ] =  = .en, for each   ,  ∘     =    =  (43) is implies each   = , that is,    = .Further, for each   ℒ , let us consider   =  ∘    for  =     (44) from  to .For each   ,   =     =   =  ∘           ∞ (45) since  is an AK-space, whence, us, we obtain  =  by the Proposition 12, whence,  has the AD-property.Also,  has the AB-property by Proposition 11.Hence,  is an AK-space by Proposition 13.
, that another de�nition of ℓ   by the complementary function  to  is  is the class of all sequences  =    such that ∑ ∞ =    < ∞ and each     * .Further, for each   ℓ  ,  .is norm is said to be Orlicz norm on ℓ  .On ℓ  , the norms  ⋅   and  ⋅   are equivalent, and   ⩽   ⩽   . * [10] converges to  in  where    and ℱ is the family of all �nite subsets of the index set  which is directed by the inclusion relation ⊆.Furthermore, {   is called a -Schauder basis for  whenever each   is continuous.uswesaythateachhas the representation By taking  =  in the �e�nition[10]we now prove that ℎ  ( has an unconditional -Schauder basis.Proof.Let us take   =   ∶ ℎ  (     ( =   as a coordinate pro�ection in the �e�nition[10].We should prove that the net (  ( ∶ ℱ ⊆ converges to  in ℎ  (.is means, for each   0, we should �nd an  0  ℱ such that ‖    (‖ ( <  for  0 ⊆ .Now, let   0 be given.Since ∑ ∞ = (‖  ‖/ < ∞ for every   0, especially for   0, the series ∑ ∞ = (‖  ‖/ is absolutely convergent and hence it is unconditional convergent in real numbers.Hence we can �nd an  0 ( such that ∑ ∞ = 0 + (‖  ‖/  .Now let  0 = { 2 …   0 .Obviously,  0 is dependent on  and the set Hence, for some   ℱ such that  0 ⊆ , we have     ( ( =    …       0    + …   2   0   2 +  …      0    + … (   and uniqueness of the sequence {   in the representation can be done similarly in the classical case.iscompletes the proof.One of our main results is the following theorem which states the generalized -dual of ℎ  ( with respect to the Banach space .e above theorem brings that ℎ  ( is an AK-space and we can use eorem 14 to �nd the generalized -dual of ℎ  (.eorem 20.Let ,  be Banach spaces and   be mutually complementary Orlicz functions.en, ℎ  (  is isomorphic by the mapping   ( ∘    to the Banach space  really de�nes a norm on   and it is a Banach space with this norm.Let   ℒ(ℎ  (  and say   = ∘  for each .is implies ‖(  ∘  (‖ = 0 so that   (   = 0 for each .Since each   ℎ  ( has the unconditional representation  = ∑ ∞ = (  ∘   (, we can write    if and only if each       so    by the de�nition of each   , that is, Ψ is one to one.Also, for an arbitrary     , if we de�ne the operator  by  (.is means the series ∑   (   is convergent, that is,  is well de�ned.�urther, that the mapping Ψ is onto, that is,   (ℎ  (  comes from the following equalities: ℒ(  since ‖  ‖ ⩽ ‖‖‖  ‖ = ‖‖.Now, let us de�ne the mapping Ψ ∶ ℒ ℎ  (   ⟶    by Ψ ( =  =    ∞ = ;   =  ∘   .isshows at the same time that Ψ is an isometry.