Some New Classes in Topological Sequence Spaces Related to $L_r$-Spaces and an Inclusion Theorem for K(X)-Spaces

Boos, Johann; Leiger, Toivo

doi:10.4171/zaa/582

Cited by 4 publications

(11 citation statements)

References 9 publications

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“…Note that S C on account of (i), and 7 0 is a subspace of F' because of (ii) and (iii). If S = S then S is called ew*_closed; furthermore, is called the Ow*_closure of S. In the special case that 0 = 0ba is the set of bounded sequences, we have (see [4]) S S and…”

Section: (Ii) Va€k V(ha)€o : (Aha)€ementioning

confidence: 99%

“…Based on the idea of L,-spaces, the notions of L w-spaces and L0-spaces were introduced by the authors in [4] (see [2] as well). In the caseof any K(Y)-space we have (see [4])…”

Section: Lp-spaces Kalton's Closed Graph Theoremmentioning

confidence: 99%

“…(b) If (F, rp) is separable and rF is an M-topology, then M fl F' = F'. (Recall that a Kspace (F, TF) containing W has W -sequentially dense dual if ' = F', see [4,Definition 3.5].) [A proof of this statement may be based on [17,Theorem 9.5.3].…”

Section: Sequentially Barrelled Mackey Spaces As Domain Spacementioning

confidence: 99%

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’Restricted’ Closed Graph Theorems

Boos¹,

Leiger²

1997

Z. Anal. Anwend.

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The paper deals with 'restricted' closed' graph theorems in the following sense: Let a suitable class E of Mackey spaces as domain spaces, a class F of locally convex spaces as range spaces and for each F E F a suitable class Cpq3 of linear maps T : E -F (E E E) with closed graph defined by a (general) property T be given. In demand is a sufficient or even a sufficient as well as necessary condition to F E F such that each T E Cf, p is continuous. Note that in the situation of the well-known closed graph theorems of Pták and Kalton, £ is the class of barrelled spaces and Mackey spaces with sequentially complete weak dual space, respectively, F is the class of all locally convex spaces and for each F E F the class Cp ip is defined to be the set of all linear maps T : E -i F (E E 1) with closed graph; a sufficient condition for F in the asked sense is B,-completeness (Ptak [10]) and to be a separable B,-complete space (Kalton [8]), respectively; a sufficient as well as necessary condition for F is A-completeness (Zhu and Zhao [20]) and to be an L,-space (Qiu [11], see [12] too), respectively. Based on the ideas developed in [4] and [2] in the present paper the notions of 8-completeness and of (8, F, q3)-spaces are introduced and a very general 'restricted' closed graph theorem of the described type (see Theorem 2.4) is proved. Furthermore, it is shown that this main result contains both new interesting special cases and a series of known closed graph theorems, for example those due to Ptik and Kalton and more generally that of Zhu and Zhao as well as that of Qiu. In addition, the paper deals with further aspects which are related to (8, F, T)-spaces and motivated by the -topologies introduced by Ruckle [13]. At last, the general considerations serve to extend the well-known inclusion theorems of Bennett and Kalton [1, Theorem 5 and 4] which establish a connection between functional analysis (weak sequential completeness and barrelledness) and summability.

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Section: (Ii) Va€k V(ha)€o : (Aha)€ementioning

confidence: 99%

“…Based on the idea of L,-spaces, the notions of L w-spaces and L0-spaces were introduced by the authors in [4] (see [2] as well). In the caseof any K(Y)-space we have (see [4])…”

Section: Lp-spaces Kalton's Closed Graph Theoremmentioning

confidence: 99%

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’Restricted’ Closed Graph Theorems

Boos¹,

Leiger²

1997

Z. Anal. Anwend.

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“…Following [5,6] is generated by a family ᏽ of semi-norms, and, moreover, let s ∈ S be a sum on S (cf. [17]), that is,…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Further and more deep-seated connections between topological properties of the dual pair (E, E β ) and the SAK -property, the continuity of matrix maps on E and the structure of domains of matrix methods have been presented, for example, in well-known inclusion theorems by Bennett and Kalton [3, Theorems 4 and 5] (see also [5,6] for generalizations).…”

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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