Definition of Vector Space

 Let 𝔽 be any field. A set V with two operation (+:V×V→V called vector addition, and •:𝔽×V→V called scalar multiplication) is a vector space over the field 𝔽 if and only if satisfy all the axioms below.

1.  ∀v,w ∈ V, v + w = +(v,w) ∈ V (V is closed under vector addition).
   
2.  ∀v,w ∈ V, v + w = w + v (Commutativity of vector addition).
3.  ∀u,v,w ∈ V, u + (v + w) = (u + v) + w (Associativity of vector addition).
4.  ∃0 ∈ V (0 is called the zero vector) such that ∀v ∈ V v + 0 = v (Existence of identity element of vector addition).
5. ∀v ∈ V, ∃-v ∈ V (-v is called the additive inverse of v) such that v + (-v) = 0 (Existence of inverse element of vector addition).
6. ∀v ∈ V, ∀a ∈ 𝔽, •(a,v) = a•v = av ∈ V (V is closed under scalar multiplication)
7. ∀v ∈ V, ∀a,b ∈ 𝔽, a(bv) = a•(b•v) = (a×b)•v = (ab)v , where × is the field multiplication of field 𝔽 (Compability of scalar multiplication with field multiplication).
8. ∀v ∈ V, 1v = v, where 1 is the multiplicative identity in 𝔽 (Identity element of scalar multiplication).
9. ∀v,w ∈ V, ∀a ∈ 𝔽, a(u + v) = au + av (Distributivity of scalar multiplication with respect to vector addition)
10. ∀v ∈ V, ∀a,b ∈ 𝔽, (a+b)v = av + bv (Distributivity of scalar multiplication with respect to field addition)