scholarly journals Dual Spaces and Hahn-Banach Theorem

2014 ◽  
Vol 22 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Extension Theorem ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Dual Spaces ◽  
Basic Properties ◽  
Banach Theorem ◽  
Real Linear

Summary In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].

scholarly journals Multilinear Operator and Its Basic Properties

2019 ◽  
Vol 27 (1) ◽  
pp. 35-45
Author(s):  
Kazuhisa Nakasho
Keyword(s):  
Algebraic Structure ◽  
Multilinear Operator ◽  
Multilinear Operators ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Basic Properties ◽  
Normed Linear Spaces ◽  
Real Linear

Summary In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure [2] of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to [3], [7], [5], [6] in this formalization.

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 A Hahn-Banach theorem for semifields

1969 ◽  
Vol 10 (1-2) ◽  
pp. 20-22 ◽  
Author(s):  
Martin Kleiber ◽  
W. J. Pervin
Keyword(s):  
Type Theorem ◽  
Linear Functionals ◽  
Linear Spaces ◽  
Banach Theorem ◽  
Real Linear

Iseki and Kasahara (see [3]) have given a Hahn-Banach type theorem for semifield-valued linear functionals on real linear spaces. We shall generalize their result by considering linear spaces over semifields.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality

2018 ◽  
Vol 11 (3) ◽  
pp. 793-802
Author(s):  
Mahdi Iranmanesh ◽  
M. Saeedi Khojasteh ◽  
M. K. Anwary
Keyword(s):  
Numerical Range ◽  
Alternative Definition ◽  
Normed Spaces ◽  
Operator Approach ◽  
Linear Spaces ◽  
Continuity Properties ◽  
Lower Semi

In this paper, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed spaces. Moreover, we investigate some of continuity properties of this kind of orthogonality. More precisely, we show that the set valued function F(x; y) = {μ : μ ∈ C, p(x − μy, y) = 1} is upper and lower semi continuous, where p(x, y) = sup{pz1,...,zn−2 (x, y) : z1, . . . , zn−2 ∈ X} and pz1,...,zn−2 (x, y) = kPx,z1,...,zn−2,yk−1 where Px,z1,...,zn−2,y denotes the projection parallel to y from X to the subspace generated by {x, z1, . . . , zn−2}. This can be considered as an alternative definition for numerical range in linear spaces.

scholarly journals Some fixed point theorems in n-normed spaces

2020 ◽  
Vol 25 (3) ◽  
pp. 1-15 ◽  
Author(s):  
Hanan Sabah Lazam ◽  
Salwa Salman Abed
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Normed Space ◽  
Fixed Point Theorems ◽  
Normed Spaces ◽  
Weak Contraction ◽  
Contraction Mappings ◽  
Basic Properties ◽  
Definition Of

In this article, we recall the definition of a real n-normed space and some basic properties. fixed point theorems for types of Kannan, Chatterge, Zamfirescu, -Weak contraction and  - (,)-Weak contraction mappings in  Banach spaces.

scholarly journals Bidual Spaces and Reflexivity of Real Normed Spaces

2014 ◽  
Vol 22 (4) ◽  
pp. 303-311
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Uniform Boundedness ◽  
Linear Operators ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Boundedness Theorem ◽  
Natural Mapping ◽  
Real Normed Space ◽  
Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

On the Intrinsic Core of Convex Cones in Real Linear Spaces

2021 ◽  
Vol 31 (2) ◽  
pp. 1276-1298
Author(s):  
Bahareh Khazayel ◽  
Ali Farajzadeh ◽  
Christian Günther ◽  
Christiane Tammer
Keyword(s):  
Convex Cones ◽  
Linear Spaces ◽  
Real Linear
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