Definition:Preordered Vector Space

Definition

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $\succeq$ be a preordering on $X$ such that:

$(1) \quad$ for all $v, v', w, w' \in X$ such that $v \succeq v'$ and $w \succeq w'$, we have $v + w \succeq v' + w'$

$(2) \quad$ for all $\alpha \in \R_{\ge 0}$ and $v, v' \in X$ with $v \succeq v'$, we have $\alpha v \succeq \alpha v'$.

We call $\struct {X, \succeq}$ a preordered vector space.

Also see

• Characterization of Preordered Vector Spaces

• Results about preordered vector spaces can be found here.

Sources

• 2023: Jean-Bernard Bru and Walter Alberto de Siqueira Pedra: C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics ... (previous) ... (next): $1.1$: Basic notions