Nuclear Fusion Physics and Technology/Algebra summary

Definition: Cartesian multiplicationEdit

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: ProjectionEdit

Projection f from set A to set B is AxB subset defined as
$f = \{ (|a>,|b>) \in A \times B : \forall |a> \in A \exists_1 |b> \in B \}$
and notation $f(|a>) = |b>$ is used.

Definition: Body of numbersEdit

Body of numbers T is defined as
$T = \{ c \in \mathbb{C}: (\exists c_1, c_2)(c_1 \neq c_2) \wedge (\forall c_1, c_2)(\exists c_3 = c_1 + c_2 \wedge \exists c_4 = c_1 . c_2 \wedge \exists c_5 = -c_1 \wedge \exists 0 \neq c_6 = c^{-1}_1) \}$

Definition: Vector spaceEdit

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

Definition: FunctionEdit

Function is a projection $f: \mathbb{C} \rightarrow \mathbb{C}$, which meets
$(\forall x \in D_f )(\exists_1 y \in H_f) (f(x) = y)$

Definition: FunctionalEdit

Functional is a projection $f: \mathbb{V} \rightarrow \mathbb{C}$, which meets
$(\forall x \in D_f )(\exists_1 y \in H_f) (f(x) = y)$

Definition: OperatorEdit

Operator is a projection $f: \mathbb{V} \rightarrow \mathbb{V}$, which meets
$(\forall x \in D_f )(\exists_1 y \in H_f) (f(x) = y)$