PhyslibAlpha.ProbabilisticTheory.Weight.Continuous

Let $E$ be an Archimedean order-unit space and let $\|\cdot\|_{\mathbf{1}}$ be its canonical order-unit norm. Given a finite weight $w : E^+ \to [0, \infty)$ defined on the positive cone of $E$, this definition constructs its canonical extension to a continuous linear functional $\tilde{w} : (E, \|\cdot\|_{\mathbf{1}}) \to \mathbb{R}$. The continuity is guaranteed by the property that $|\tilde{w}(x)| \le w(\mathbf{1}) \|x\|_{\mathbf{1}}$ for all $x \in E$.

Let $E$ be an Archimedean order-unit space and let $w$ be a finite weight on its positive cone $\text{PosCone}(E)$. Let $\tilde{w}: (E, \|\cdot\|_{\mathbf{1}}) \to \mathbb{R}$ be the continuous linear map associated with $w$. For any element $x$ in the positive cone $\text{PosCone}(E)$, the value of $\tilde{w}(x)$ is equal to the real-valued evaluation of the weight $w(x)$.