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.

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 Orderability of topological spaces by continuous preferences

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Linear Orders ◽  
Mathematical Economics

We extend van Dalen and Wattel's (1973) characterization of orderable spaces and their subspaces by obtaining analogous results for two larger classes of topological spaces. This type of spaces are defined by considering preferences instead of linear orders in the former definitions, and possess topological properties similar to those of (totally) orderable spaces (cf. Alcantud, 1999). Our study provides particular consequences of relevance in mathematical economics; in particular, a condition equivalent to the existence of a continuous preference on a topological space is obtained.

A Note on Extending Locally Finite Collections

1976 ◽  
Vol 19 (1) ◽  
pp. 117-119
Author(s):  
H. L. Shapiro ◽  
F. A. Smith
Keyword(s):  
Topological Space ◽  
Set Cover ◽  
Topological Spaces ◽  
Locally Finite ◽  
Whole Space

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

scholarly journals Some fixed-point theorems on locally convex linear topological spaces

1975 ◽  
Vol 13 (2) ◽  
pp. 241-254 ◽  
Author(s):  
E. Tarafdar
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Topological Spaces ◽  
The Family ◽  
Image Position ◽  
Commutative Family ◽  
Locally Convex

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the setcontains a point of M. Then the family F has a common fixed-point in M.

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 Bornologiese pseudotopologiese vektorruimtes

1990 ◽  
Vol 9 (1) ◽  
pp. 15-18
Author(s):  
M. A. Muller
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Direct Sums ◽  
Paper Properties ◽  
Topological Vector Spaces ◽  
Quotient Spaces ◽  
Locally Convex

Homological spaces were defined by Hogbe-Nlend in 1971 and pseudo-topological spaces by Fischer in 1959. In this paper properties of bornological pseudo-topological vector spaces are investigated. A characterization of such spaces is obtained and it is shown that quotient spaces and direct sums o f boruological pseudo-topological vector spaces are bornological. Every bornological locally convex pseudo-topological vector space is shown to be the inductive limit in the category of pseudo-topological vector spaces of a family of locally convex topological vector spaces.

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