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.

Linear Normed Spaces with Extension Property

1966 ◽  
Vol 9 (4) ◽  
pp. 433-441 ◽  
Author(s):  
George Elliott ◽  
Israel Halperin
Keyword(s):  
Linear Space ◽  
Linear Extension ◽  
Normed Linear Space ◽  
Extension Property ◽  
Linear Mapping ◽  
Normed Spaces ◽  
Bounded Linear ◽  
The Real ◽  
Real Normed Linear Space

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).

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

Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range

2003 ◽  
Vol 46 (2) ◽  
pp. 216-228 ◽  
Author(s):  
Chi-Kwong Li ◽  
Leiba Rodman ◽  
Peter Šemrl
Keyword(s):  
Linear Space ◽  
Numerical Range ◽  
Positive Definiteness ◽  
Linear Operators ◽  
Bounded Linear ◽  
Real Linear Space ◽  
The Real ◽  
Selfadjoint Operators ◽  
Linear Maps ◽  
Real Linear

AbstractLet H be a complex Hilbert space, and be the real linear space of bounded selfadjoint operators on H. We study linear maps ϕ: → leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps ϕ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption ϕ has the form , for a suitable invertible or unitary T and ξ ∈ {1, −1}, where Xt stands for the transpose of X relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on H. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form αI + K, where α is a scalar and K is compact.

scholarly journals Operators with a Single Spectrum

1966 ◽  
Vol 15 (1) ◽  
pp. 11-18 ◽  
Author(s):  
T. T. West
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Operators ◽  
Complex Field ◽  
Unique Extension ◽  
Single Spectrum ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Resolvent Set ◽  
Image Position

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.

New Fixed Point Theorems via Contraction Mappings in Complete Intuitionistic Fuzzy Normed Linear Space

2018 ◽  
Vol 15 (01) ◽  
pp. 65-83
Author(s):  
Nabanita Konwar ◽  
Ayhan Esi ◽  
Pradip Debnath
Keyword(s):  
Fixed Point ◽  
Mathematical Models ◽  
Linear Space ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Existence Of A Solution ◽  
Linear Spaces ◽  
Contraction Mappings ◽  
Normed Linear Spaces ◽  
Intuitionistic Fuzzy

Contraction mappings provide us with one of the major sources of fixed point theorems. In many mathematical models, the existence of a solution may often be described by the existence of a fixed point for a suitable map. Therefore, study of such mappings and fixed point results becomes well motivated in the setting of intuitionistic fuzzy normed linear spaces (IFNLSs) as well. In this paper, we define some new contraction mappings and establish fixed point theorems in a complete IFNLS. Our results unify and generalize several classical results existing in the literature.

Gaps between Spheres in Normed Linear Spaces

1980 ◽  
Vol 23 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Infinite Dimensional ◽  
Normed Linear Spaces ◽  
Concentric Spheres ◽  
General Conditions

AbstractThe geometric notions of a gap and gap points between “concentric” spheres in a normed linear space are introduced and studied. The existence of gap points characterizes finitedimensional spaces. General conditions are given under which an infinite-dimensional normed linear space admits concentric spheres such that both these spheres and their dual spheres fail to have gap points.

On the calculation of the Dunkl-Williams constant of normed linear spaces

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Hiroyasu Mizuguchi ◽  
Kichi-Suke Saito ◽  
Ryotaro Tanaka
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Normal Structure ◽  
Linear Spaces ◽  
Normed Linear Spaces

AbstractRecently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.

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
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