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 $$f = \{ (|a>,|b>) \in A \times B : \forall |a> \in A \exists_1 |b> \in B \} $$ and notation $$f(|a>) = |b> $$ is used.

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

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

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

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