PhyslibAlpha.ProbabilisticTheory.OrderUnit.Bidual

The unit element $1$ in the bidual space $\text{Bidual}(E)$ of an order-unit space $E$ is defined as the canonical image of the unit $1 \in E$. For any positive functional $\phi$, the value of this bidual unit is the total weight $\phi(1)$.

Let $E$ be an order-unit space. For any positive functional $\phi$ on $E$, the unit element $1$ of the bidual space $\text{Bidual}(E)$ satisfies $(1)(\phi) = \phi(1)$, where $\phi(1)$ is the total weight of the functional.

Let $E$ be an order-unit space and let $\text{ofE} : E \to \text{Bidual}(E)$ be the canonical embedding. The image of the order unit $1 \in E$ under this embedding is equal to the unit element $1$ of the bidual space $\text{Bidual}(E)$.

For any element $x$ in the bidual of a system $E$, there exists a natural number $n$ such that for every positive functional $\phi$ on $E$, the absolute value of the value assigned by $x$ to $\phi$ is bounded by $n$ times the total weight $\phi(1)$. That is, $|x(\phi)| \leq n \phi(1)$.

The bidual space $\text{Bidual}(E)$ of an order-unit space $E$ is itself an order-unit space. The order unit is defined as the canonical image of the unit $1 \in E$, such that for any element $x \in \text{Bidual}(E)$, there exists a natural number $n$ where $x \le n \cdot 1$. This structure also ensures that the unit is non-negative.

The bidual space $\text{Bidual}(E)$ of an order-unit space $E$ is an Archimedean order-unit space. This means that the bidual inherits the Archimedean property: for any element $x \in \text{Bidual}(E)$, if $x \le \epsilon \cdot 1$ for all real $\epsilon > 0$, then $x \le 0$, where $1$ is the unit of the bidual defined by the total weight of functionals $\phi \mapsto \phi(1)$.

Let $E$ be an order-unit space and $\text{Bidual}(E)$ be its bidual space. If an element $x$ of the bidual satisfies $x \leq 1$ (where $1$ is the bidual unit), then for any positive linear functional $\phi: E \to \mathbb{R}$, the value of $x$ at $\phi$ is at most the total weight of the functional, i.e., $x(\phi) \leq \phi(1)$.

Let $E$ be an order-unit space. If the dual cone of $E$ (the set of positive linear functionals $\phi: E \to \mathbb{R}$) forms a lattice—a property defining a classical system—then the bidual space $\text{Bidual}(E)$ is an order-unit lattice. This means that $\text{Bidual}(E)$ is an Archimedean order-unit space that also possesses a lattice structure (joins and meets) compatible with its ordering.