scholarly journals A new class of fractional type set-valued functions

2019 ◽  
Vol 35 (1) ◽  
pp. 79-84
Author(s):  
ALEXANDRU ORZAN ◽  
Keyword(s):  
Special Class ◽  
Convex Sets ◽  
Linear Spaces ◽  
New Class ◽  
Finite Dimensional ◽  
Affine Functions ◽  
Real Linear ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Fractional Type

The so-called ratios of affine functions, introduced by Rothblum (1985) in the framework of finite-dimensional Euclidean spaces, represent a special class of fractional type vector-valued functions, which transform convex sets into convex sets. The aim of this paper is to show that a similar convexity preserving property holds within a new class of fractional type set-valued functions, acting between any real linear spaces.

SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

2015 ◽  
Vol 99 (2) ◽  
pp. 145-165 ◽  
Author(s):  
G. BEER ◽  
J. VANDERWERFF
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Separation Theorems ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Weakly Closed ◽  
Separation Of Convex Sets ◽  
Real Linear ◽  
Closed Convex Set ◽  
Special Case

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

scholarly journals Semiconvex geometry

1981 ◽  
Vol 30 (4) ◽  
pp. 496-510 ◽  
Author(s):  
Joe Flood
Keyword(s):  
Algebraic Variety ◽  
Convex Sets ◽  
Convex Geometry ◽  
Algebraic Approach ◽  
Algebraic Proof ◽  
Linear Spaces ◽  
Structural Decomposition ◽  
Banach Theorem ◽  
Real Linear ◽  
Convex Subsets

AbstractSemiconvex sets are objects in the algebraic variety generated by convex subsets of real linear spaces. It is shown that the fundamental notions of convex geometry may be derived from an entirely algebraic approach, and that conceptual advantages result from applying notions derived from algebra, such as ideals, to convex sets. Some structural decomposition results for semiconvex sets are obtained. An algebraic proof of the algebraic Hahn-Banach theorem is presented.

scholarly journals Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

2010 ◽  
Vol 18 (4) ◽  
pp. 207-212 ◽  
Author(s):  
Takao Inoué ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Vector Valued Functions ◽  
Vector Valued

Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).

Porosity and ε‎-Fr échet differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Porous Set ◽  
Delta Set ◽  
Lipschitz Maps ◽  
Finite Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Null Set

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.

scholarly journals Convex sets in proximal relator spaces

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3411-3414 ◽  
Author(s):  
J.F. Peters
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Convex Sets ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Finite Dimensional

This article introduces convex sets in finite-dimensional normed linear spaces equipped with a proximal relator. A proximal relator is a nonvoid family of proximity relations R? (called a proximal relator) on a nonempty set. A normed linear space endowed with R? is an extension of the Sz?z relator space. This leads to a basis for the study of the nearness of convex sets in proximal linear spaces.

scholarly journals Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

2011 ◽  
Vol 19 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Takao Inoué ◽  
Adam Naumowicz ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Linear Spaces ◽  
Partial Differentiation ◽  
Normed Linear Spaces ◽  
Vector Valued Functions ◽  
Vector Valued

UNIQUENESS OF THE TOPOLOGY ON SPACES OF VECTOR-VALUED FUNCTIONS

2001 ◽  
Vol 64 (2) ◽  
pp. 445-456 ◽  
Author(s):  
A. R. VILLENA
Keyword(s):  
Banach Space ◽  
Linear Subspace ◽  
Locally Compact Group ◽  
Translation Invariant ◽  
Linear Spaces ◽  
Space Topology ◽  
Isolated Points ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Topological Linear

Let Ω be a topological space without isolated points, let E be a topological linear space which is continuously embedded into a product of countably boundedly generated topological linear spaces, and let X be a linear subspace of C(Ω, E). If a ∈ C(Ω) is not constant on any open subset of Ω and aX ⊂ X, then it is shown that there is at most one F-space topology on X that makes the multiplication by a continuous. Furthermore, if [Ufr ] is a subset of C(Ω) which separates strongly the points of Ω and [Ufr ]X ⊂ X, then it is proved that there is at most one F-space topology on X that makes the multiplication by a continuous for each a ∈ [Ufr ].These results are applied to the study of the uniqueness of the F-space topology and the continuity of translation invariant operators on the Banach space L1(G, E) for a noncompact locally compact group G and a Banach space E. Furthermore, the problems of the uniqueness of the F-algebra topology and the continuity of epimorphisms and derivations on F-algebras and some algebras of vector-valued functions are considered.

Sign in / Sign up

Export Citation Format

Share Document