The Goldstine theorem for asymmetric normed linear spaces (Q837642)

If \(X\) is a linear space, then a function \(q: X\to\mathbb{R}^+\) is called an asymmetric norm on \(X\) if for all \(x,y\in X\) and \(r\in \mathbb{R}^+\), \(x= 0\) if and only if \(q(x)= q(-x)= 0\), \(q(rx)= rq(x)\) and \(q(x+ y)= q(x)+ q(y)\). It follows that the function \(q^s: X\to\mathbb{R}\) defined by \(d^s(x)= \max(q(x), q(-x))\) is a norm on \(X\). The authors study the dual and bidual spaces of \((X,q)\) and of \((X,q^s)\) and establish a characterization of reflexive asymmetric normed linear spaces.