Complex Numbers form Vector Space over Themselves

Theorem
The set of complex numbers $\C$, with the operations of addition and multiplication, forms a vector space.

Proof
Let the field of complex numbers be denoted $\struct {\C, +, \times}$.

By Complex Numbers under Addition form Abelian Group, $\struct {\C, +}$ is an abelian group.

From Complex Multiplication Distributes over Addition:

From Complex Multiplication is Associative:
 * $\forall x, y, z \in \C: x \times \paren {y \times z} = \paren {x \times y} \times z$

From Complex Multiplication Identity is One
 * $\forall x \in \C: 1 \times x = x$

Therefore $\struct {\C, +, \times}$ forms a vector space.

Also see

 * Properties of Complex Numbers