Nuclear Fusion Physics and Technology/Algebra summary

Definition: Cartesian multiplication

Cartesian multiplication of two sets A, B with |a>,|b> elements is defined as
${\displaystyle A\times B=\{(|a>,|b>):|a>\in A\wedge |b>\in B\}}$

Definition: Projection

Projection f from set A to set B is AxB subset defined as
${\displaystyle f=\{(|a>,|b>)\in A\times B:\forall |a>\in A\exists _{1}|b>\in B\}}$
and notation ${\displaystyle f(|a>)=|b>}$ is used.

Definition: Body of numbers

Body of numbers T is defined as
${\displaystyle T=\{c\in \mathbb {C} :(\exists c_{1},c_{2})(c_{1}\neq c_{2})\wedge (\forall c_{1},c_{2})(\exists c_{3}=c_{1}+c_{2}\wedge \exists c_{4}=c_{1}.c_{2}\wedge \exists c_{5}=-c_{1}\wedge \exists 0\neq c_{6}=c_{1}^{-1})\}}$

Definition: Vector space

Vector space V is defined as
${\displaystyle \mathbb {V} =\{(V,T,f_{1},f_{2}):(\forall |a>,|b>\in V)(f_{1}(|a>,|b>)=f_{1}(|b>,|a>))(...)\}}$

Definition: Function

Function is a projection ${\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} }$, which meets
${\displaystyle (\forall x\in D_{f})(\exists _{1}y\in H_{f})(f(x)=y)}$

Definition: Functional

Functional is a projection ${\displaystyle f:\mathbb {V} \rightarrow \mathbb {C} }$, which meets
${\displaystyle (\forall x\in D_{f})(\exists _{1}y\in H_{f})(f(x)=y)}$

Definition: Operator

Operator is a projection ${\displaystyle f:\mathbb {V} \rightarrow \mathbb {V} }$, which meets
${\displaystyle (\forall x\in D_{f})(\exists _{1}y\in H_{f})(f(x)=y)}$