On the extension of positive linear functionals (Q1974404)

Let \(L\) be a subspace of real linear space \(E\). Let \(K\) be a subset of \(E\) such that \(K+ K\subset K\), \(\alpha K\subset K\) for ll \(\alpha\geq 0\). A monotonic Banach functional with respect to \(K\) is a mapping \(\omega:E\to \mathbb{R}\) so that: (1) \(\omega(x+ y)\leq \omega(x)+ \omega(y)\), \(x,y\in E\). (2) \(\omega(\lambda x)= \lambda\omega(x)\), \(x\in E\), \(\lambda\geq 0\). (3) \(\omega(x+ y)\geq \omega(x)\), \(x\in E\), \(y\in K\). The authors obtain the following theorem: Let \(f_0\) be a linear functional on \(L\) so that \(f_0(x)\geq 0\) for every \(x\in K\cap L\). Then a necessary and sufficient condition for the existence of a linear functional \(f: E\to \mathbb{R}\) such that \(f_0(x)= f(x)\) for every \(x\in L\) and \(f(x)\geq 0\) for every \(x\in K\), is the existence of a monotonic Banach functional \(\omega\) with respect to \(K\) satisfying the inequality \(f_0(x)\leq \omega(x)\) for every \(x\in L\). Other results of extension of positive linear functionals are given.