🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

A vector space over a field $F$ is a non-empty set $V$ with a binary operation $\oplus$ on $V$ and a binary function $\otimes :F\times V\to V$ such that the following hold for any $a,b\in F$ and $u,v\in V$

• $\oplus$ is associative and commutative
• There is a identity element $0_V$ with respect to $\oplus$
• $1_F$ is an identity element for $\otimes$
• $\oplus$ has inverses for $0_V$
• $\otimes$ distributes into $\oplus$
• $\left(a{+}_{F}b\right)\otimes v=(a\otimes v)\oplus (b\otimes v)$
• $a\otimes (b\otimes v)=\left(a{·}_{F}b\right)\otimes v$