The Dual Space of an Asymmetric Normed Linear Space

On balancedness and D-completeness of the space of semi-Lipschitz functions

“…Let us recall [1,6] that if (X, p) is a quasi-normed linear space then the socalled dual algebraic of (X, p) is the cone X * consisting of all linear real-valued functions on X that are upper semicontinuous on ( X, τ (d p )…”

“…Note that, in this case, the function q −1 : X → R + given by q −1 (x) = q(−x) is also a quasi-norm on X and the function q s : X → R + given by q s (x) = max q(x), q(−x) is a norm on X. As in [6], we say that (X, q) is a biBanach space if (X, q s ) is a Banach space.…”

“…Let us also recall that p * is the function from X * to R + dened by p * (f ) = sup f (x) : p(x) 1 for all f ∈ X * [1,6], and thus (X * , p * ) is a normed cone which is said to be the dual space of (X, p). …”

Continuous operators on asymmetric normed spaces

“…A function p : X → R + is an asymmetric norm on X ( [13], [15]) if for all x, y ∈ X and r ∈ R + , (i) p(x) = p(−x) = 0 if and only if x = 0.…”

Asymmetric Norms and the dual complexity spaces.

“…If (X, q) is an extended asymmetric normed linear space such that the induced extended quasi-metric d q is bicomplete, we will say, as in the usual case, that (X, q) is a biBanach space ( [21], [24], [46]). …”