Page "Linear subspace" Paragraph 15

Firstly, property 1 ensures W is nonempty.

Looking at the definition of a vector space, we see that properties 2 and 3 above assure closure of W under addition and scalar multiplication, so the vector space operations are well defined.

Since elements of W are necessarily elements of V, axioms 1, 2 and 5-8 of a vector space are satisfied.

By the closure of W under scalar multiplication ( specifically by 0 and-1 ), axioms 3 and 4 of a vector space are satisfied.