Absolute convexity in certain topological linear spaces

1969 ◽  
Vol 66 (3) ◽  
pp. 541-545 ◽  
Author(s):  
I. J. Maddox ◽  
J. W. Roles
Keyword(s):  
Linear Space ◽  
Topological Linear Space ◽  
Linear Spaces ◽  
Empty Subset ◽  
Image Position ◽  
Topological Linear

For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U andIt is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).

scholarly journals AN ANALOGY OF HAHN–BANACH SEPARATION THEOREM FOR NEARLY TOPOLOGICAL LINEAR SPACES

2021 ◽  
Vol 7 (1) ◽  
pp. 81
Author(s):  
Madhu Ram
Keyword(s):  
Linear Space ◽  
Separation Theorem ◽  
Topological Linear Space ◽  
Alternative Definition ◽  
Linear Spaces ◽  
Definition Of ◽  
Topological Linear

In this paper, we introduce the notion of nearly topological linear spaces and use it to formulate an alternative definition of the Hahn–Banach separation theorem. We also give an example of a topological linear space to which the result is not valid. It is shown that \(\mathbb{R}\) with its ordinary topology is not a nearly topological linear space.

scholarly journals Monotone mappings in topological linear spaces

1965 ◽  
Vol 5 (1) ◽  
pp. 25-35
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Real Number ◽  
Linear Space ◽  
Number Field ◽  
Open Subset ◽  
Zero Element ◽  
Monotone Mappings ◽  
Topological Linear Space ◽  
Linear Spaces ◽  
Topological Linear ◽  
Real Number Field

Let E be a topological linear space over the real number field. Throughout of this paper, we denote by G an open subset of E, by ∂G the boundary of G and by the closure of G. The totality of all circled open neighbourhoods of the zero element denoted by U.

On the convex kernel of a compact set

1967 ◽  
Vol 63 (2) ◽  
pp. 311-313 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Linear Space ◽  
Compact Subset ◽  
Line Segment ◽  
Linear Dimension ◽  
Single Point ◽  
Compact Set ◽  
Topological Linear Space ◽  
Topological Linear

Suppose that E is a compact subset of a topological linear space ℒ. Then the convex kernel K, of E, is such that a point k belongs to K if every point of E can be seen, via E, from k. Valentine (l) has asked for conditions on E which ensure that the convex kernel K, of E, consists of exactly one point, and in this note we give such a condition. If A, B, C are three subsets of E, we use (A, B, C) to denote the set of those points of E, which can be seen, via E, from a triad of points a, b, c, with a ∈ A, b ∈ B, c ∈ C. We shall say that E has the property if, whenever A is a line segment and B, C are points of E which are not collinear with any point of A, the set (A, B, C) has linear dimension of at most one, and degenerates to a single point whenever A is a point.

scholarly journals Nearly directed subspaces of partially ordered linear spaces

1968 ◽  
Vol 16 (2) ◽  
pp. 135-144
Author(s):  
G. J. O. Jameson
Keyword(s):  
Linear Space ◽  
Transitive Relation ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Partially Ordered ◽  
Real Linear ◽  
Image Position ◽  
Ordered Linear Spaces

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

scholarly journals On Exposed and Farthest Points in Normed Linear Spaces

1971 ◽  
Vol 12 (3) ◽  
pp. 301-308 ◽  
Author(s):  
M. Edelstein ◽  
J. E. Lewis
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Nonempty Subset ◽  
Linear Spaces ◽  
Farthest Points ◽  
Normed Linear Spaces ◽  
Farthest Point ◽  
Image Position

Let S be a nonempty subset of a normed linear space E. A point s0 of S is called a farthest point if for some x ∈ E, . The set of all farthest points of S will be denoted far (S). If S is compact, the continuity of distance from a point x of E implies that far (S) is nonempty.

scholarly journals Matrix transformations and convex spaces

1988 ◽  
Vol 11 (3) ◽  
pp. 585-588
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Matrix Transformations ◽  
Topological Linear Space ◽  
Sequential Completeness ◽  
Metrizable Spaces ◽  
Topological Linear ◽  
Convex Spaces

In a Hausdorff topological linear space we examine relations betweenr-convexity and a condition on matrix transformations between null sequences. In particular, for metrizable spaces the condition impliesr-convexity. For locally bounded spaces the condition implies sequential completeness.

C(X) As A Dual Space

1972 ◽  
Vol 24 (3) ◽  
pp. 485-491 ◽  
Author(s):  
E. G. Manes
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Dual Space ◽  
Minkowski Functional ◽  
Topological Linear Space ◽  
Open Set ◽  
Topological Linear

It is known [1] that for compact Hausdorff X, C(X) is the dual of a Banach space if and only if X is hyperstonian, that is the closure of an open set in X is again open and the carriers of normal measures in C(X)* have dense union in X. With the desiratum of proving that C(X) is always the dual of some sort of space we broaden the concept of Banach space as follows. A Banach space may be comfortably regarded as a pair (E, B) where E is a topological linear space and B is a subset of E ; the requisite property is that the Minkowski functional of B be a complete norm whose topology coincides with that of E.

A Tauberian theorem for statistical convergence

1989 ◽  
Vol 106 (2) ◽  
pp. 277-280 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Tauberian Theorem ◽  
Statistical Convergence ◽  
Slow Oscillation ◽  
Topological Linear Space ◽  
Tauberian Condition ◽  
Topological Linear ◽  
Locally Convex

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

A Note on Best Simultaneous Approximation in Normed Linear Spaces

1976 ◽  
Vol 19 (3) ◽  
pp. 359-360 ◽  
Author(s):  
Arne Brøndsted
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Convex Functions ◽  
Existence And Uniqueness ◽  
Simultaneous Approximation ◽  
Normed Linear Space ◽  
Present Note ◽  
Linear Spaces ◽  
Best Simultaneous Approximation ◽  
Image Position

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that

Sign in / Sign up

Export Citation Format

Share Document