An Abstract Concept of the Sum of a Numerical Series

1970 ◽  
Vol 22 (4) ◽  
pp. 863-874 ◽  
Author(s):  
William H. Ruckle
Keyword(s):  
Topological Space ◽  
Sequence Space ◽  
Dual Space ◽  
Linear Topological Space ◽  
Sequence Spaces ◽  
Abstract Concept ◽  
Numerical Series ◽  
Locally Convex ◽  
Theory Of Duality

Our aim in this paper, generally stated, is to formulate an abstract concept of the notion of the sum of a numerical series. More particularly, it is a study of the type of sequence space called “sum space”. The idea of sum space arose in connection with two distinct problems.1.1 The Köthe-Toeplitz dual of a sequence space T consists of all sequences t such that st ∈ l1 (absolutely summable sequences) for each s∈T. It is known that if cs or bs is used in place of l1, an analogous theory of duality for sequence spaces can be developed (cf. [2]). What other spaces of sequences can play a rôle analogous to l1? This problem is treated in [6].1.2. Let {xn, fn} be a complete biorthogonal sequence in (X, X*), where X is a locally convex linear topological space and X* is its topological dual space.

scholarly journals A characterization of pre-near-standardness in locally convex linear topological spaces

1972 ◽  
Vol 6 (1) ◽  
pp. 107-115
Author(s):  
J.J.M. Chadwick ◽  
R.W. Cross
Keyword(s):  
Topological Space ◽  
Dual Space ◽  
Linear Topological Space ◽  
Topological Spaces ◽  
Locally Convex

Let X be a locally convex linear topological space. A point z in an ultralimit enlargement of X is pre-near-standard if and only it is finite and for every equicontinuous subset S′ of the dual space X′, a point z′ belongs to *S′ ∩ μσ(X′, X) (0) only if z′ (z) is infinitesimal.

scholarly journals Dual pairs of sequence spaces

2001 ◽  
Vol 28 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Johann Boos ◽  
Toivo Leiger
Keyword(s):  
Sequence Space ◽  
Dual Pair ◽  
General Concept ◽  
Sequence Spaces ◽  
Dual Pairs ◽  
The Third ◽  
Convex Sequence ◽  
Starting Point ◽  
Alpha Beta ◽  
Locally Convex

The paper aims to develop for sequence spacesEa general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz dualsE×(×∈{α,β})combined with dualities(E,G),G⊂E×, and theSAK-property (weak sectional convergence). TakingEβ:={(yk)∈ω:=𝕜ℕ|(ykxk)∈cs}=:Ecs, wherecsdenotes the set of all summable sequences, as a starting point, then we get a general substitute ofEcsby replacingcsby any locally convex sequence spaceSwith sums∈S′(in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair(E,ES)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,Eβ). 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 Boos and Leiger (1993 and 1997)), 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.

Some criteria for nuclearity

1986 ◽  
Vol 100 (1) ◽  
pp. 151-159 ◽  
Author(s):  
M. A. Sofi
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequence Spaces ◽  
Bilinear Forms ◽  
Simple Formulation ◽  
Normal Topology ◽  
Nuclear Spaces ◽  
Locally Convex

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

scholarly journals Fixed point theorems for condensing multivalued mappings on a locally convex topological space

1975 ◽  
Vol 12 (2) ◽  
pp. 161-170 ◽  
Author(s):  
E. Tarafdar ◽  
R. Výborný
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theorems ◽  
Linear Topological Space ◽  
General Definition ◽  
Multivalued Mappings ◽  
Locally Convex

A general definition for a measure of nonprecompactness for bounded subsets of a locally convex linear topological space is given. Fixed point theorems for condensing multivalued mappings have been proved. These fixed point theorems are further generalizations of Kakutani's fixed point theorems.

A Decomposition Theorem for m-Convex Sets in Rd

1976 ◽  
Vol 28 (5) ◽  
pp. 1051-1057
Author(s):  
Marilyn Breen
Keyword(s):  
Topological Space ◽  
Convex Sets ◽  
Decomposition Theorem ◽  
Linear Topological Space ◽  
Local Convexity ◽  
Line Segments ◽  
Local Nonconvexity ◽  
Locally Convex

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of S, at least one of the line segments determined by these points lies in S. A point x in S is said to be a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z Є N ⌒ S, then [y, z] ⊆ S. If S fails to be locally convex at some point a in S, then q is called a point of local nonconvexity (lnc point) of S.

A Generalization of the Mapping Degree

1974 ◽  
Vol 26 (5) ◽  
pp. 1109-1117 ◽  
Author(s):  
D. G. Bourgin
Keyword(s):  
Topological Space ◽  
Linear Topological Space ◽  
Dimensional Case ◽  
Lefschetz Number ◽  
Mapping Degree ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Locally Convex

For the single-valued case the notion of degree has been given recent expression by papers of Dold [5] for the finite dimensional case, and by Leray-Schauder [8] for the locally convex linear topological space. Klee [7] has removed this restriction by use of shrinkable in place of convex neighborhoods with the central role filled by a form of (2.15) below. For set-valued maps a modern formulation is, for instance, to be found in Gorniewicz-Granas [6]. These contributions relate the degree to the Lefschetz number, and the set-valued maps are required to map points into acyclic sets; that is to say, into "swollen points".

Intersections of m-Convex Sets

1975 ◽  
Vol 27 (6) ◽  
pp. 1384-1391 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Topological Space ◽  
Convex Sets ◽  
Linear Topological Space ◽  
Local Convexity ◽  
Line Segments ◽  
Local Nonconvexity ◽  
Locally Convex

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of line segments determined by these points lies in S. A point x in S is called a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z ∈ N⋂ S, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S.

scholarly journals Linear functionals on Orlicz sequence spaces without local convexity

1992 ◽  
Vol 15 (2) ◽  
pp. 241-254 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Local Convexity ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Orlicz Sequence Space ◽  
Orlicz Sequence Spaces ◽  
Locally Convex

The general form of continuous linear functionals on an Orlicz sequence space1ϕ(non-separable and non-locally convex in general) is obtained. It is proved that the spacehϕis anM-ideal in1ϕ.

scholarly journals Onβ-dual of vector-valued sequence spaces of Maddox

2002 ◽  
Vol 30 (7) ◽  
pp. 383-392 ◽  
Author(s):  
Suthep Suantai ◽  
Winate Sanhan
Keyword(s):  
Sequence Space ◽  
Dual Space ◽  
Sequence Spaces ◽  
Bk Space ◽  
Vector Valued

Theβ-dual of a vector-valued sequence space is defined and studied. We show that if anX-valued sequence spaceEis a BK-space having AK property, then the dual space ofEand itsβ-dual are isometrically isomorphic. We also give characterizations ofβ-dual of vector-valued sequence spaces of Maddoxℓ(X,p),ℓ∞(X,p),c0(X,p), andc(X,p).

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