Let

V be a vector space.Then,

1.

V is closed under vector addition.(The sum of two vectors is again a vector.)

2.For any two vectors

and

in

V we have by definition,

(Commutativity)3.For any three

,

and

in

V,we again have by definition

(Associativity)4.There exists a null vector such that,

for all

in

V.

(Existence of identity)5.There exists a 'negative' vector

for any

in

V such that,

(Existence of inverse)Thus,

V satisfies all the requirements of an abelian group.