Nuclear Fusion Physics and Technology/Algebra summary

Definition: Cartesian multiplication

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

Definition: Projection

Projection f from set A to set B is AxB subset defined as
$f=\{(|a>,|b>)\in A\times B:\mathrm{\forall }|a>\in A{\mathrm{\exists }}_1|b>\in B\}$
and notation $f(|a>)=|b>$ is used.

Definition: Body of numbers

Body of numbers T is defined as
$T=\{c\in ℂ:(\mathrm{\exists }{c}_1,c_2)(c_1\ne c_2)\wedge (\mathrm{\forall }{c}_1,c_2)(\mathrm{\exists }{c}_3=c_1+c_2\wedge \mathrm{\exists }{c}_4=c_1.c_2\wedge \mathrm{\exists }{c}_5=-c_1\wedge \mathrm{\exists }0\ne c_6=c_1^{-1})\}$

Definition: Vector space

Vector space V is defined as
$𝕍=\{(V,T,f_1,f_2):(\mathrm{\forall }|a>,|b>\in V)(f_1(|a>,|b>)=f_1(|b>,|a>))(...)\}$ Template:BookCat

Definition: Function

Function is a projection $f:ℂ\to ℂ$, which meets
$(\mathrm{\forall }x\in D_f)({\mathrm{\exists }}_{1}y\in H_f)(f(x)=y)$

Definition: Functional

Functional is a projection $f:𝕍\to ℂ$, which meets
$(\mathrm{\forall }x\in D_f)({\mathrm{\exists }}_{1}y\in H_f)(f(x)=y)$

Definition: Operator

Operator is a projection $f:𝕍\to 𝕍$, which meets
$(\mathrm{\forall }x\in D_f)({\mathrm{\exists }}_{1}y\in H_f)(f(x)=y)$