Vector Spaces

A vector space $\mathcal{V}$ over a field $\mathbb{F}$ is a set of elements (vectors) {|$v$⟩} closed under the action of two maps:

• $+:\mathcal{V}\times \mathcal{V}\to \mathcal{V}$ (Commutative, Associative, Identity $(|0⟩)$, Inverse $(-|\cdot ⟩)$ )
• $\ast :\mathbb{F}\times \mathcal{V}\to \mathcal{V}$ (Compatibility, Distributivity)

$\mathbb{F}$ can be $\mathbb{R},\mathbb{C}$ or others.