Wang and Hao Journal of Inequalities and Applications 2012, 2012:71 http://www.journalofinequalitiesandapplications.com/content/2012/1/71

RESEARCH Open Access Full invariants on certain class of abstract dual systems Fubin Wang1* and Cuixia Hao2

* Correspondence: Abstract [email protected] 1Department of , In this article, we study the convergence of sequences of operators on some classical Heilongjiang College of vector-valued sequence spaces in the frame of abstract dual systems. Several Construction, Harbin 150025, China Full list of author information is consequences of full invariants are obtained. available at the end of the article Keywords: sequential-evaluation convergence, abstract , full invariant, uniformly exhaustive, essentially bounded

1 Introduction A dual system in linear analysis consists of a linear space and a collection of linear functionals defined on the linear space. Theory of locally convex spaces is exactly about theory of dual systems which plays a crucial role in many fields of mathematical analysis. In fact, many people working in different fields of mathematics have been devoting themselves on the research of some special dual systems such as measure sys- tem (Σ,Ca(Σ,X)), abstract function system (Ω,C(Ω,X)) and operator system (X,L(X,Y)) as well as fuzzy system (U,F(U)) etc. In recent years, a few consequences have enlarged the known invariant ranges of boundedness, subseries convergence, and sequential-evaluation convergence etc., and theory of spaces has achieved substantial development. It is no doubtful that invariant principles make great influence on the trend of different fields of mathematical analysis. This article is motivated by the problems considered by the authors in [1-7]. And the purpose of this article is to study full invariant of the convergence in p p (Yl (X), lp(X))-topology on the dual system (Yl (X), lp(X)),(0

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It is not difficult to know that not every admissible polar topology b(X,Y) is a compa- tible topology on X wrt the dual system (X, Y). Definition 1.3 [9]. If a property P of X is shared by all admissible polar topologies lying between s(X,Y) and b(X,Y), then P is called a full invariant. E We denote (E, F) an abstract dual system, where E =0, F ⊂ X and X is a locally convex space. F A topology τ on E is called a (E, F)-topology if there exists ℱ ⊂ 2 such that ∪M Î ℱM=F and τ is a topology of on each set M Î ℱ,whichmeans thatasequence{un} ⊂ (E,τ)issaidtoconvergetoapointu Î (E,τ) if and only if for every M Î ℱ,{f(un)} converges uniformly to f(u)forallf Î M [10]. We use s(E,F)to represent the weakest (E,F)-topology, i.e., in the topological space (E,s(E,F)), a sequence {un} ⊂ (E,s(E,F)) converges to a point u Î (E, s(E, F)) if and only if for every f Î F,{f(un)} converges to f(u). Similarly, for an abstract dual system (F,E), we can also give the concept of (F, E)- topology, denoted by τ*(F, E). The weakest (F, E)-topology is denoted by s*(F, E), called weak star topology. For the convenience, the sequence {fn}convergestof in (F, s* F, E F, τ* F, E σ ∗(F,E) τ ∗(F,E) ( )) and in ( ( )) is denoted by fn −−−−→ f and fn −−−−→ f, respectively. In F, E b* F,E β∗(F,E) addition, ( )-topology ( ) is called strong star topology if fn −−−−→ f implies that fn ® f for all (F, E)-topology, say τ*(F, E)onF. Obviously, b*(F, E) is the strongest (F, E)-topology on F. Full invariant of (F, E) are analog to those of (E,F).

p 2 Full invariant on the dual system (Yl (X), lp(X)) (0

X p p Y and Yl (X) represent the sets of Y - valued operators with domains X and l (X), respectively. Let ⎧ ⎫ ⎨  ⎬ p βY ⊂ X ∈ p l (X) = ⎩(Aj) Y : Aj(xj) is convergent for all (xj) l (X)⎭ , j

p bY p l (X) is called the b- of l (X) [4]. A Î lp X bY x Î lp X p → ∈ Suppose that ( j) ( ) , for each ( j) ( ), we define f(Aj),n : l (X) Y, n N p → and f(Aj) : l (X) Y by     

f(Aj),n xj = Aj xj , 1≤j≤n

and     

f(Aj) xj = Aj xj . j Wang and Hao Journal of Inequalities and Applications 2012, 2012:71 Page 3 of 8 http://www.journalofinequalitiesandapplications.com/content/2012/1/71

{ }⊂ lp (X) ∈ lp(X) Obviously, f(Aj),n Y , f(Aj) Y and   p σ ∗ Yl (X),lp(X) −−−−−−−−→ ( →∞) f(Aj),n f(Aj) n

p in the dual system (Yl (X), lp(X)). p In the following, we shall first characterize a collection of sets in l (X) generating ∗ p strong star topology β (Yl (X), lp(X)).    p  p Definition 2.1 [2]. If M ⊂ l (X) and lim ≥ xj = 0 uniformly for (xj) Î M,that n→∞ j n is, for all ε >0, there exists jε Î N such that     p sup xj ≤ ε, ∈ (xj) M j≥jε

then M is called uniformly exhaustive set. p p Denote M[l (X)] = {M ⊂ l (X):M is uniformly exhaustive}. p p bY Lemma 2.1 [2]. Let M ⊂ l (X), 0

Proof Necessity. Otherwise, we suppose that

σ ∗ lp(X) p −−−−−−−−(Y ,l (X→)) ( →∞) f(Aj),n f(Aj) n

but   p τ ∗ Yl (X),lp(X) −−−−−−−−→ f(Aj),n f(Aj)

p p doesn’t hold. By the definition of l (X) p -topology, there must exist M Î ℳ[l   (Y   , l (X)) (X)] such that lim f( ) xj = f( ) xj not uniformly for all (xj) Î M, i.e., the series n Aj ,n Aj ∑jAj(xj) is not uniformly convergent with respect to all (xj) Î M. Thus, there is ε >0

such that for any m0 Î N, there exist m>m0 and (xj) Î M satisfying          Aj(xj) ≥ 2ε.   j≥m

p bY Since (Aj) Î l (X) , Σj Aj (xj) is convergent and there exists n>msuch that           Aj(xj) <ε.   j≥n+1 Wang and Hao Journal of Inequalities and Applications 2012, 2012:71 Page 4 of 8 http://www.journalofinequalitiesandapplications.com/content/2012/1/71

Hence                    Aj(xj) =  Aj(xj) − Aj(xj)  ≤ ≤   ≥ ≥  m j n j m  j n+1               ≥  Aj(xj) −  Aj(xj)     j≥m j≥n+1 > 2ε − ε = ε.

Since M is uniformly exhaustive, there exists j1 Î N such that for every (xj) Î M,    1 x p < . j 2 j≥j1

Thus, there exist integers n1 >m1 >j1 and (x1j) Î M such that           Aj(x1j) >ε.   m1≤j≤n1

Again using the uniform exhaustion of M, we can get j2 >n1 such that for every (xj) Î M,    1 x p < . j 22 j≥j2

So there exist n2 >m2 >j2 and (x2j) Î M such that           Aj(x2j) >ε.   m2≤j≤n2

In the same procedure, we can get a sequence of integers m1

and           Aj(xkj) >ε, k =1,2,...   mk≤j≤nk

Let  x , m ≤ j ≤ n , k =1,2,..., y = kj k k j 0, otherwise.

p Then (yj) Î l (X). In fact,         1 y p = x p < = 1 j kj 2k j k mk≤j≤nk k Wang and Hao Journal of Inequalities and Applications 2012, 2012:71 Page 5 of 8 http://www.journalofinequalitiesandapplications.com/content/2012/1/71

However                    Aj(yj) =  Aj(xkj) >ε, k =1,2,....     mk≤j≤nk mk≤j≤nk

p bY This is a contradiction with (Aj) Î l (X) . Sufficiency. It is trivial, we omit it. p Remark. For any (Yl (X), lp(X))-topology, let N [lp(X)] be a collection of sets generat- ing the topology. Then by Lemma 2.1 we can show that for any N ∈ N [lp(X)], N is ∗ p uniformly exhaustive, that is, N [lp(X)] ⊂ M[lp(X)].Soτ (Yl (X), lp(X))-topology is ∗ p exactly the strongest β (Yl (X), lp(X))-topology. Therefore, the following consequence of Theorem 2.1 is immediate. p ∗ p Corollary 2.1.Let(Yl (X), lp(X)) be a dual system, τ (Yl (X), lp(X)) a p p p p bY (Yl (X), lp(X))-topology on Yl (X) generated by ℳ[l (X)],(Aj) Î l (X) ,then τ ∗ lp(X) p −−−−−−−−(Y ,l (X→)) →∞ if and only if f(Aj),n f(Aj)(n )

β∗ lp(X) p −−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj)(n ).

∗ p The Corollary 2.1 shows that the convergence of τ (Yl (X), lp(X)) is equivalent to ∗ p that of β (Yl (X), lp(X)). Combining this with Theorem 2.1, we obtain the next conse- { } quence which means that the convergence of f(Aj),n is full invariant. p ∗ p Theorem 2.2.Let(Yl (X), lp(X)) be a dual system, τ (Yl (X), lp(X)) be a lp(X) p p p p bY (Y , l (X))-topology on Yl (X) generated by ℳ[l (X)], (Aj) Î l (X) ,then σ ∗ lp(X) p −−−−−−−−(Y ,l (X→)) →∞ if and only if f(Aj),n f(Aj)(n )

β∗ lp(X) p −−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj)(n ).

∞ ∞ 3 Full invariant on the dual system (Yl (X), l (X)) N Let X be the set of all X-valued sequences,     ∞ N   l (X)= (xj) ∈ X :sup xj < +∞ j∈N

X ∞ ∞ and Y , Yl (X) represent the sets of Y-valued operators with domains X and l (X), respectively. Let ⎧ ⎫ ⎨  ⎬ ∞ βY ⊂ X ∈ ∞ l (X) = ⎩(Aj) Y : Aj(xj) is convergent for all (xj) l (X)⎭ , j

∞ bY ∞ then l (X) is called the b-dual space of l (X) [4]. A Î l∞ X bY x Î l∞ X ∞ → n Î Suppose that ( j) ( ) ,forevery( j) ( ), we define f(Aj),n : l (X) Y, N ∞ → and f(Aj) : l (X) Yby Wang and Hao Journal of Inequalities and Applications 2012, 2012:71 Page 6 of 8 http://www.journalofinequalitiesandapplications.com/content/2012/1/71

  

f(Aj),n xj = Aj(xj) 1≤j≤n

and   

f(Aj) xj = Aj(xj). j   ⊂ l∞ (X) ∈ l∞(X) Obviously, f(Aj),n Y , f(Aj) Y , and

∞ σ ∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj)(n )

∞ in the dual system (Yl (X), l∞(X)). ∞ Next we shall first find a collection of sets in l (X) generating strong star topology ∞ ∞ β∗ l (X) ∞ β∗(Yl (X), l∞(X)). And then we prove that −−−−−−−−−(Y ,l (X→)) →∞.Further- f(Aj),n f(Aj) (n ) ∞ ∞ X → →∞ l (X) l ( ) more, we specify that f(Aj),n f(Aj)(n ) (Y , )-topology admit full invariant. ∞ Definition 3.1 [2]. M ⊂ l (X) is called essential bounded set if there is j0 Î N such that     sup xj < ∞ (xj)∈M,j≥jo

∞ ∞ Denote ℳ[l (X)] = {M ⊂ l (X):M is essential bounded}. ∞ ∞ bY Lemma 3.1 [2]. Let X and Y be Banach spaces, M ⊂ l (X), (Aj) Î l (X) .If∑j Aj(xj) is uniformly convergent for all (xj) Î M, then M is essential bounded. ∞ ∞ ∗ ∞ ∞ Theorem 3.1.Let(Yl (X), l (X)) be a dual system, τ (Yl (X), l (X)) be a ∞ ∞ ∞ ∞ ∞ bY (Yl (X), l (X))-topology on Yl (X) generated by ℳ[l (X)], (Aj) Î l (X) ,then ∞ σ ∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ if and only if f(Aj),n f(Aj)(n )

∞ τ ∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj)(n ).

Proof. Necessity. Otherwise, we suppose that

∞ σ ∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj)(n )

but

∞ τ ∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj)(n )

∞ doesn’t hold. Then there must exist M Î ℳ[l (X)] such that

lim f(A ),n[(xj)] = f(A )[(xj)] xj Î M ∑j Aj xj n j j not uniformly for all ( ) , i.e., the series ( )isnot uniformly convergent with respect to (xj) Î M. Thus, there exist ε >0,asequenceof

integers m1

Let  x , m ≤ j ≤ n , k =1,2,..., y = kj k k j 0, otherwise.

Since M is essential bounded, there is j0 Î N such that         sup yj ≤ sup zj < +∞. ≥ j j0 (zj)∈M,j≥j0

∞ Then (yj) Î l (X). But we have                    Aj(yi) =  Aj(xkj) ≥ ε, k =1,2,....     mk≤j≤nk mk≤j≤nk

∞ bY This is a contradiction with (Aj) Î l (X) . Necessary. Trivial. The Lemma 3.1 leads us to the following consequences. ∞ ∞ ∗ ∞ ∞ ∞ Corollary 3.1.Let(Yl (X), l (X)) be a dual system, τ (Yl (X), l (X)) bea(Yl (X), ∞ ∞ ∞ ∞ bY l (X))-topology on Yl (X) generated by ℳ[l (X)], (Aj) Î l (X) ,then ∞ τ ∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ if and only if f(Aj),n f(Aj)(n ).

∞ β∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj) (n )

∞ ∞ ∗ ∞ ∞ ∞ Theorem 3.2.Let(Yl (X), l (X)) be a dual system, τ (Yl (X), l (X)) be a (Yl (X), ∞ ∞ ∞ ∞ bY l (X))-topology on Yl (X) generated by ℳ[l (X)], (Aj) Î l (X) ,then ∞ σ ∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ if and only if f(Aj),n f(Aj)(n )

∞ β∗ l (X) ∞ −−−−−−−−−(Y ,l (X→)) →∞ f(Aj),n f(Aj)(n ).

4 Conclusion In a frame of some certain class of dual system, we discuss a few problems on conver- gence of the sequences of operators in classical vector-valued sequence spaces and we obtain some consequences on full invariants. Precisely, sequences of operators in the p ∞ spaces Yl (X) and Yl (X) are studied, respectively. We not only find the strongest sequential-evaluation convergence but also obtain a series of theorems about full invar- iants on the sequential evaluation convergence.

Acknowledgements We would like to express our deep gratitude to the referee for meaningful advice which makes a great improvement of the original manuscript, and gave many thanks to professor Shusen Ding for his careful suggestions. FW was greatly supported by Heilongjiang College of Construction. CH was supported by a grant from Ministry of Education of Heilongjiang Province supporting overseas returned scholars (1055HZ003), and by a grant from Ministry of Education ofHeilongjiang Province (11551366).

Author details 1Department of Mathematics, Heilongjiang College of Construction, Harbin 150025, China 2Department of Mathematics, Heilongjiang University, Harbin 150080, China

Authors’ contributions FW conceived of the study. FW and CH participated in the design of the proof and drafted the manuscript. All authors read and approved the final manuscript. Wang and Hao Journal of Inequalities and Applications 2012, 2012:71 Page 8 of 8 http://www.journalofinequalitiesandapplications.com/content/2012/1/71

Competing interests The authors declare that they have no competing interests.

Received: 26 October 2011 Accepted: 26 March 2012 Published: 26 March 2012

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doi:10.1186/1029-242X-2012-71 Cite this article as: Wang and Hao: Full invariants on certain class of abstract dual systems. Journal of Inequalities and Applications 2012 2012:71.

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