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International Journal of Analysis
Volume 2013 (2013), Article ID 862949, 7 pages
Generalized K####^~^~^~^~^~^####xf6;the-Toeplitz Duals of Some Vector-Valued Sequence Spaces
Department of Mathematics, Inonu University, 44280 Malatya, Turkey
Received 17 August 2012; Accepted 26 November 2012
Academic Editor: Chuanxi Qian
Copyright © 2013 Y####^~^~^~^~^~^####x131;lmaz Y####^~^~^~^~^~^####x131;lmaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a scalar-valued FK-space originated by the AK-property. Our main interest in this work is to expose relationships between the operator space and and the generalized -duals of some -valued AK-space where and are Banach spaces and . Further, by these results, we obtain the generalized -duals of some vector-valued Orlicz sequence spaces.
The idea of dual sequence space was introduced by K####^~^~^~^~^~^####xf6;the and Toeplitz . Then, Maddox, , generalized this notion to -valued sequence classes where is a Banach space. This brings an important contribution to the operator matrix transformation of Banach space-valued sequence spaces. Remember that - and -duals of a (complex-valued) sequence space , denoted by and , respectively, are defined to be where is the space of all complex-valued sequences. The in the classical definitions of K####^~^~^~^~^~^####xf6;the-Toeplitz duals is replaced by a sequence of linear operators, not necessarily continuous, from into another Banach space . Thus, if is a nonempty set of sequences with , then generalized - and -duals of are defined to be respectively. It is clear that this notion depends on the space and if , then where is the space of all linear operators from into . Without the loss of generality we can restrict ourselves in this work to continuous operators and being the space of all continuous linear operators from into and being the space of all -valued sequences which is a natural generalization of .
We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a sequence space whenever it has the AK-property. Related results are also expressed in [3, page 176]. Here, we are going to show that there is an analogue relationship for -valued sequence spaces in the context of generalized -duals with respect to another fixed Banach space . Further, by applying this result, we obtain generalized -duals of some vector-valued Orlicz sequence spaces. We think that our results give a fruitful way to find generalized duals of this kind of vector-valued sequence spaces.
We use the notations , and 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 Fr####^~^~^~^~^~^####xe9;chet 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, and so its topology is the weakest topology such that the projections 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 Then possesses the same properties as the function . Suppose now Then is an Orlicz function. The functions and are called mutually complementary Orlicz functions, and they satisfy the Young inequality,
An Orlicz function is said to satisfy the -condition for small at if for each there exist and such that , for all .
The space consists of all sequences of scalars such that and it becomes a Banach space which is called an Orlicz sequence space with the Luxemburg norm The space is closely related to the space which is an Orlicz sequence space with .
Another definition of , , is given by the complementary function to as follows: where is the complementary function to and####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; is the collection of all in with . Clearly,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; and are normed by the Orlicz norm It was shown that these two norms on are equivalent.
An important closed subspace of , introduced by Y. Garibanov, is which is defined by
Immediately, we can introduce the vector-valued extension of the spaces and for any Banach space . Therefore, where is the space of all -valued sequences and is the norm of . is a Banach space with the Luxemburg norm and it coincides with whenever . Further, define the closed subspace of by if and only if If satisfies the -condition then .
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 . Thus, the classical Hahn-Banach theorem can be restated in the following way for operators (see also ).
Theorem 1. Let , be Banach spaces and let be a continuous linear operator of rank 1. Then is a Hahn-Banach operator.
By some modification on the assertion: norm preserving, the result remains true if is taken as a locally convex space. The evidence of this assertion can be found in [6, Section 7.2.].
Hence, by using Theorem 1 we derived some tools for later sections as in the way that is similar to classical treatments. The proof of the following result is also given in . Nevertheless, it will be convenient to restate it here.
Corollary 2. Let be Banach spaces and . Then, for some , there exists a corresponding operator such that
Proof. If is not null, take . Then is a closed subspace of hence is a Banach space with the same norm. Define , for some , from into . Then satisfies the required condition of Theorem 1 on . Hence it is a Hahn-Banach operator such that . The norm preserving extension of to has the desired properties. The result is obvious for .
Corollary 3. Let be an lc space, be a Banach space, be a vector subspace of , and . Then there exists an and a corresponding operator such that
Proof. Let and consider . Fix some and define the operator for (equivalently, for some and ( or ) such that ). Clearly, the hyphothesis says that is not dense in . Thus, must be closed in since it is a maximal subspace of (see [6, Prob. ]). But , hence is continuous. Further, is an operator of rank 1 such that and on . Thus, the extension of is the desired operator.
Let us establish an lc topology on a Banach space with respect to another Banach space . 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 field of then coincide with the usual weak topology. It is clear that, a net which is -convergent to is also weak convergent to . The converse of this assertion is not true.
Example 4. Let . Then the sequence of unit vectors is weak convergent to in [8, page 99]. But, it is not -convergent to . Therefore, for the identity operator on , we have .
However, we cannot work this example in (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 definition of the Schur property.
Theorem 5. Let be a Banach space having the Schur property. Then weak convergence implies -convergence in for every Banach space .
Now consider the canonical embedding , where is the space of all continuous operators from into and and are Banach spaces, which assigns each to the operator on defined by Clearly, so that and Theorem 1 and the succeeding corollary assert that the canonical embedding is a linear isometry from into 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. The converse of this assertion is the following theorem.
Theorem 6. Let and be Banach spaces. Then -bounded sets are norm bounded.
Proof. Let be -bounded and be canonical embedding of into . 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 for each .
Theorem 7. Let be an lc space and be a Banach space. Then -bounded sets are also bounded in the lc topology of .
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. Thus we can show as in Theorem 6 that is bounded in , whence, is bounded in , that is, is a bounded subset of .
We conclude this section with a brief discussion of equicontinuity. A set of linear maps from one topological vector space into another one is called equicontinuous if, for each neighborhood of in , is a neighborhood of in . Equicontinuity is a generalization of the uniform boundedness of the family of linear maps between seminormed spaces.
Theorem 8 (see ). Let be a collection of continuous linear mappings from the Fr####^~^~^~^~^~^####xe9;chet space into the topological vector space . Then is equicontinuous if and only if the set is bounded in , for each .
Lemma 9 (see ). Let be a net of continuous operators from a Fr####^~^~^~^~^~^####xe9;chet space into the topological vector space . Then the set is a closed subspace of .
4. Sectional Properties and Operator Spaces
For an , is called th section of . Further, denotes the space of all finite sequences in .
Definition 10. 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.
An AK-(AD-, AB-) space is also called to have AK-(AD-, AB-) property.
Let be an FK-space and define the set Clearly, and whenever is an AB-space.
To define another important classes we consider the mappings and define the set , for some Banach space , by
Proposition 11. For each Banach space .
Proof. Let and then we can write , that is, Since ; that is, the sequence is -convergent hence it is -bounded. Thus, it is also bounded in the lc Fr####^~^~^~^~^~^####xe9;chet topology of by Theorem 7.
Proposition 12. For each Banach space , where is the closure of in .
Proof. Let . Then, for every such that on , This implies . If this is not so, then there exists an and a corresponding operator such that by Corollary 3. This is a contradiction.
Proposition 13. Let be an FK-space with AD- and AB-property. Then also has the AK-property.
Proof. Define Then the sequence is bounded by the AB-property. Therefore is equicontinuous by Theorem 8. On the other hand for each , That is, Since is a closed subspace of from Lemma 9, we obtain that . Thus for each ( by the AD-property), whence, has the AK-property.
Theorem 14. Let be an FK-space. Then is an AK-space if and only if and are isomorphic for every Banach space .
Proof. Let where each and define by
for each . Write
where is the th (continuous) projection defined by . Then each is continuous and the sequence is pointwise convergent since the series is convergent. So, the operator , which is also defined by
is continuous by the Banach-Steinhauss closure theorem, whence, . That is injective comes from the following discussion. Let . Then, for each ,
This implies each , that is, . Further, for each , let us consider
from to . For each ,
since is an AK-space, whence, . This means that is surjective.
Conversely, let and be isomorphic. Then each has the representation such that each and also, for each , This shows that , that is, Thus, 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.
5. Applications on Vector-Valued Orlicz Sequence Spaces
It is not hard to see as in the classical case, , that another definition of by the complementary function to is where is the class of all sequences such that and each . Further, for each , defines a norm on . This norm is said to be Orlicz norm on .
Lemma 15. On , the norms and are equivalent, and .
Proofs of this lemma and the above assertion can be given in a similar way followed in [4, Theorem 8.9], by using the inequality and by using the fact that if and only if .
Lemma 16. Let be an Orlicz function. The sets are identical.
Proof. Let , this means for . Hence, , that is, . Conversely, let , that is This means for some . Therefore since is nondecreasing.
In general has no Schauder basis in classical manner. In  we introduce a new kind basis notion. Let us give this definition and prove that has a basis in this manner.
Definition 17 (see ). Let and be Banach spaces and be a set. A family of continuous linear functions is called -basis for if the following condition is satisfied. There exists a directed subset (by some relation ) of 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 converges to in where for each and is the family of all finite subsets of the index set which is directed by the inclusion relation . Furthermore, is called a Y-Schauder basis for whenever each is continuous.
Thus we say that each has the representation in this case.
Definition 18. The -basis in the above definition is called unconditional whenever with the inclusion relation .
By taking in the Definition  we now prove that has an unconditional -Schauder basis.
Theorem 19. For consider again the operators such that Then, the sequence is an unconditional -Schauder basis for .
Proof. Let us take as a coordinate projection in the Definition . We should prove that the net converges to in . This means, for each , we should find an such that for . Now, let be given. Since for every , especially for , the series is absolutely convergent and hence it is unconditional convergent in real numbers. Hence we can find an such that . Now let . Obviously, is dependent on and the set includes the . This means Now, remember that Hence, for some such that , we have The continuity of each and uniqueness of the sequence in the representation can be done similarly in the classical case. This completes the proof.
One of our main results is the following theorem which states the generalized -dual of with respect to the Banach space . The above theorem brings that is an AK-space and we can use Theorem 14 to find the generalized -dual of .
Theorem 20. Let , be Banach spaces and be mutually complementary Orlicz functions. Then, is isomorphic by the mapping to the Banach space where each is defined as in Theorem 19.
Proof. We prove that is isometrically isomorphic to .
A routine calculation shows that really defines a norm on and it is a Banach space with this norm. Let and say for each . This implies so that for each . Since each has the unconditional representation , we can write Immediately each since . Now, let us define the mapping if and only if each so by the definition of each , that is, is one to one. Also, for an arbitrary , if we define the operator by on then, by using the Young inequality, we have Since for each and from [4, Prop. 8.12], we have Also since . This means the series is convergent, that is, is well defined. Further, that the mapping is onto, that is, comes from the following equalities: This shows at the same time that is an isometry.
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