ON THE UNIQUENESS OF EXTENSION AND UNIQUE BEST APPROXIMATION IN THE DUAL OF AN ASYMMETRIC NORMED LINEAR SPACE

. A well known result of R. R. Phelps (1960) asserts that in order that every linear continuous functional, deﬁned on a subspace Y of a real normed space X , have a unique norm preserving extension it is necessary and suﬃcient that its annihilator Y ⊥ be a Chebyshevian subspace of X ∗ . The aim of this note is to show that this result holds also in the case of spaces with asymmetric norm.


INTRODUCTION
Let X be a real linear space.A function •| : X → [0, ∞) is called an asymmetric norm if it satisfies all the usual axioms of a norm, excepting the absolute homogeneity, which is replaced by positive homogeneity, i.e., λx| = λ x| , ∀x ∈ X, ∀λ ≥ 0.
The asymmetric norm is called with extended values if there exists x ∈ X such that x| = +∞.The pair (X, •|) is called a space with asymmetric norm (see [8], [2]).
In a space with asymmetric norm it is possible that x| = −x| for some x ∈ X.The asymmetric norm generates a topology having as a neighborhood base the balls B (x, r) = {y ∈ X : y − x| < r}, x ∈ X, r ≥ 0, but the topological space X, τ •| is not a linear topological space, because the multiplication by scalars is not a continuous operation (see [2, p. 199]).
Let X # be the algebraic dual of X, i.e. the space of all linear functional on X.We say that f It is immediate that if f, g ∈ X # are bounded then their sum f + g and the product λf for λ ≥ 0 are bounded too.This shows that the set of all bounded linear functionals on X is a cone (see [2]), or an ac-space, according to [7].
In general, it is possible that for a bounded linear functional f on (X, •|) the linear functional −f be not bounded.Such an example is given by the functional f (x) = x (1) defined on the space For a bounded linear functional f put It follows that the function •| defined by ( 2) satisfies the axioms of an asymmetric norm on the cone of all bounded linear functionals on X (see [2]).
Observe that This shows that a bounded linear functional is always continuous with respect to the topology generated by the symmetric norm x = max { x| , −x|} associated to an asymmetric norm •| .
If both f and −f are bounded then the linear functional f is continuous with respect to the topology generated by the asymmetric norm.
According to [7], the set is the (symmetric) dual of (X, •|).In other words, X * s is the usual topological algebraic dual of the normed space (X, • ), where x = max { x| , −x|} .
Observe that X * is a cone in the linear space X * s .Equip the linear space X * s with the extended asymmetric norm whose restriction to the asymmetric dual X * is It is important to remark the fact that a linear functional f belongs to X * if and only if it is an upper semicontinuous linear functional on (X, •|), and that X * = f ∈ X * s : f | * s < +∞ (see [7]).
Let (Y, •|) be a subspace of the space with asymmetric norm (X, •|), and let Y * and Y * s be the dual cone and the (symmetric) dual of Y , respectively.
The following Hahn-Banach type theorem holds: Theorem 1.Let (X, •|) be a real space with asymmetric norm and (Y, •|) a subspace of it.Then for every f ∈ Y * there exists F ∈ X * such that The functional p is convex, positively homogeneous and f (y) ≤ f | * • y|, for every y ∈ Y .By the Hahn-Banach extension theorem ([8, p. 484]) there exists a linear functional F : On the other hand the set of all extensions that preserve the asymmetric norm.
By Theorem 1, the set E (f ) is always nonempty.The problem of finding necessary or/and sufficient conditions in order that every f ∈ Y * have a unique norm preserving extension is closely related to a best approximation problem in the space X * s equipped with the asymmetric norm F | * s = sup F (x) : x| ≤ 1 .
Concerning the following notions, in the case of usual spaces, one can consult Singer's book [16].
Let (Y, •|) be subspace of (X, •|) and let The set of all best approximation elements for F in Y ⊥ is denoted by The annihilator Y ⊥ of Y is a proximinal subspace of X * s and, for every F ∈ X * we have and linear continuous is called the asymmetric dual of the space with asymmetric norm (X, •|) .If x = max { x| , −x|} is the norm generated by •| on X and R is equipped with the usual absolute-value norm |•| , then the set (4)