Linear space

A space $$V$$ with defined operations:

vector addition: $$V \times V \rigtarrow V$$ (or $$v + u$$)
scalar multiplication: $$\mathbb{F} \times V \rightarrow V$$ (or $$\alpha v$$)

Axioms

associativity
commutativity
identity element of vector addition
inverse element
compatibility
identity element of scalar multiplication
distributivity of scalar multiplication with respect to field addition
distributivity of scalar multiplication with respect to vector addition