Hello!

Because `H+K sub V,` the operations (addition and multiplication by a number) are defined, and their results are in `V.` The only thing we need to prove is that the results of this operations remain in `H+K.`

**1**. Multiplication by a number. Let `a in RR` (or `CC` ) and `w in H+K.` Then by the definition of `H+K` there are `u in H` and `v in K` such that `w = u+v.` So

`a*w = a*(u+v) = a*u+a*v.`

Because H and K are subspaces, `a*u in H` and `a*v in K,` thus `a*w in H+K.`

**2**. Addition. Let `w_1, w_2 in H+K.` Then there are `u_1,u_2 in H` and `v_1,v_2 in K` such that `w_1=u_1+v_1` and `w_2=u_2+v_2.` Then

`w_1+w_2 = (u_1+v_1)+(u_2+v_2) = (u_1+u_2)+(v_1+v_2).`

Because `H` and `K` are subspaces, `u_1+u_2 in H` and `v_1+v_2 in K,` thus `w_1+w_2 in H+K.`