Definition:Preordering Induced by Convex Cone

Definition

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

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

Let $P \subseteq X$ be a convex cone in $X$.

Define a relation $\succeq^P$ by:

$v \succeq^P v'$ if and only if $v - v' \in P$

for each $v, v' \in X$.

We say that $\succeq^P$ is the preordering on $X$ induced by $P$.

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