Quotient subspaces of asymmetric normed linear spaces (Q1016748)

A function \(p:X\to\mathbb{R}^+\) on a real linear space \(X\) is an asymmetric norm on \(X\) if for all \(x,y\in X\) and \(r\in\mathbb{R}^+\) (i) \(p(x)=p(-x)=0\) if and only if \(x=0\). (ii) \(p(rx)=rp(x)\). (iii) \(p(x+y)\leq p(x)+p(y)\). The pair \((X,p)\) is called an asymmetric normed linear space. The authors show that if \(H\subseteq X\) is a linear subspace of \((X,p)\), then the function \(\widehat p:X/H\to\mathbb{R}\) on the quotient space \(X/H\) defined by \(\widehat p(\widehat x):= \inf\{p(y):y \in\widehat x\}=\inf\{p(x+h):h\in H\}\) is positive, subadditive and positively homogeneous.