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

On the Hahn-Banach Extension Property

1970 ◽  
Vol 13 (1) ◽  
pp. 9-13
Author(s):  
Ting-On To
Keyword(s):  
Linear Function ◽  
Linear Space ◽  
Linear Extension ◽  
Normed Linear Space ◽  
Extension Property ◽  
Bounded Linear ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Real Linear

In this paper, we consider real linear spaces. By (V:‖ ‖) we mean a normed (real) linear space V with norm ‖ ‖. By the statement "V has the (Y, X) norm preserving (Hahn-Banach) extension property" we mean the following: Y is a subspace of the normed linear space X, V is a normed linear space, and any bounded linear function f: Y → V has a linear extension F: X → V such that ‖F‖ = ‖f‖. By the statement "V has the unrestricted norm preserving (Hahn-Banach) extension property" we mean that V has the (Y, X) norm preserving extension property for all Y and X with Y ⊂ X.

Quasi-regular semilattices in singular linear spaces

2016 ◽  
Vol 08 (01) ◽  
pp. 1650005
Author(s):  
Baohuan Zhang ◽  
Yujun Liu ◽  
Zengti Li
Keyword(s):  
Finite Field ◽  
Linear Space ◽  
Fixed Integer ◽  
Linear Spaces ◽  
Partially Ordered ◽  
Type Formula ◽  
Singular Linear Spaces

Let [Formula: see text] denote the [Formula: see text]-dimensional singular linear space over a finite field [Formula: see text]. For a fixed integer [Formula: see text], denote by [Formula: see text] the set of all subspaces of type [Formula: see text], where [Formula: see text]. Partially ordered by ordinary inclusion, one family of quasi-regular semilattices is obtained. Moreover, we compute its all parameters.

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.

About the Polarity in a Real Linear Space

1977 ◽  
Vol 77 (1) ◽  
pp. 181-185 ◽  
Author(s):  
Jacques Bair
Keyword(s):  
Linear Space ◽  
Real Linear Space ◽  
Real Linear

scholarly journals Affine Independence in Vector Spaces

2010 ◽  
Vol 18 (1) ◽  
pp. 87-93 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Linear Space ◽  
Vector Spaces ◽  
Barycentric Coordinates ◽  
Independent Subset ◽  
Linear Combinations ◽  
Affine Hull ◽  
Real Linear Space ◽  
Real Linear ◽  
The Given

Affine Independence in Vector Spaces In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set. Finally, we introduce and prove selected properties of the barycentric coordinates.

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