Strong Boundedness and Strong Convergence in Sequence Spaces

Buntinas, Martin; Tanović-Miller, N.

doi:10.4153/cjm-1991-053-8

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…In Section 4 we examine when E S and E SS have the SB-property. In Section 5 we generalize the wellknown factorization theorem due to Garling [8, Theorem 3(a)] and deduce from this result some well-known theorems due to Buntinas [5], Buntinas and Tanović-Miller [6], Fleming [7], and Sember [16].…”

Section: Introductionmentioning

confidence: 91%

“…Moreover, we have |S| = [bs] := {z ∈ bs | sup j 2 j |z k | < ∞}. By Buntinas and Tanović-Miller[6], an FK-space E is said to have the [AB]-property, if E is an AB-space (see (3.1)) and2 n α k x k e k α = (α k ) ∈ χ, n ∈ N is bounded in E for all x ∈ E,where χ is the set of all sequences of 0's and 1's. We are going to verify that E has the [AB]-property if and only if it is an SB-space with respect to S = [cs] and s(z) := k z k .For each f ∈ E and x ∈ E from the [AB]-property follows,…”

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confidence: 96%

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Dual pairs of sequence spaces. III

Boos

Leiger

2006

Journal of Mathematical Analysis and Applications

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In the previous papers [J. Boos, T. Leiger, Dual pairs of sequence spaces, Int. J. Math. Math. Sci. 28 (2001) 9-23; J. Boos, T. Leiger, Dual pairs of sequence spaces. II, Proc. Estonian Acad. Sci. Phys. Math. 51 (2002) 3-17], the authors defined and investigated dual pairs (E, E S ), where E is a sequence space, S is a BK-space on which a sum s is defined in the sense of Ruckle [W.H. Ruckle, Sequence Spaces, Pitman Advanced Publishing Program, Boston, 1981], and E S is the space of all factor sequences from E into S. In generalization of the SAK-property (weak sectional convergence) in the case of the dual pair (E, E β ), the SK-property was introduced and studied. In this note we consider factor sequence spaces E |S| , where |S| is the linear span of B S τ ω , the closure of the unit ball of S in the FK-space ω of all scalar sequences. An FK-space E such that E |S| includes the f -dual E f is said to have the SB-property. Our aim is to demonstrate, that in the duality (E, E S ), the SB-property plays the same role as the AB-property in the case E S = E β . In particular, we show for FK-spaces, in which the subspace of all finitely nonzero sequences is dense, that the SB-property implies the SK-property. Moreover, in the context of the SB-property, a generalization of the well-known factorization theorem due to Garling [D.J.H. Garling, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967) 997-1019] is given.

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Section: Introductionmentioning

confidence: 91%

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confidence: 96%

Dual pairs of sequence spaces. III

Boos

Leiger

2006

Journal of Mathematical Analysis and Applications

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“…It is our primary aim to study these properties along the lines of previous investigations by Garling [10], Buntinas [3][4][5][6][7], Sember [21,22], Fleming [9], and many others on related sectional properties. We shall encounter many similarities but also noteworthy differences.…”

Section: Introductionmentioning

confidence: 99%

On l1-Invariant Sequence Spaces

Grosse‐Erdmann

2001

Journal of Mathematical Analysis and Applications

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We introduce and investigate a new sectional property for topological sequence spaces, which we call the KB property. It turns out that multipliers into l ∞ and multiplication by l 1 play a very similar role for KB as multipliers into bs and multiplication by bv 0 do for the well-known sectional property AB. However, we also encounter significant differences. One of these differences motivates the study of l 1 -invariant sequence spaces, that is, sequence spaces E with l 1 E = E.  2001 Academic Press

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“…That complex of problems is also connected with the USAK -property of sequence spaces (cf. Sember [19], Sember and Raphael [18] as well as Swartz [20], Swartz and Stuart [21]), in particular properties of the duality (E, E α ) where E α denotes the α-dual of E; moreover, Buntinas and Tanović-Miller [9] investigated the strong SAK -property of FK -spaces.…”

Section: Introductionmentioning

confidence: 99%

Dual pairs of sequence spaces

Boos

Leiger

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. The paper aims to develop for sequence spaces E a general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz duals E × (× ∈ {α, β}) combined with dualities (E, G), G ⊂ E × , and the SAKproperty (weak sectional convergence). TakingE cs , where cs denotes the set of all summable sequences, as a starting point, then we get a general substitute of E cs by replacing cs by any locally convex sequence space S with sum s ∈ S (in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair (E, E S ) of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality (E, E β ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Leiger (1993 and), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.2000 Mathematics Subject Classification. 46A45, 46A20, 46A30, 40A05.

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Strong Boundedness and Strong Convergence in Sequence Spaces

Cited by 5 publications

References 9 publications

Dual pairs of sequence spaces. III

Dual pairs of sequence spaces. III

On l1-Invariant Sequence Spaces

Dual pairs of sequence spaces

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