由于1.(a1,b1)+(a2,b2)=(a1+a2,b1+b2+a1a2)=(a2+a1,b2+b1+a2a1)=(a2,b2)+(a1,b1),
2.[(a1,b1)+(a2,b2)]+(a3+b3)=((a1+a2)+a3,(b1+b2)+b3+(a1a2)a3)
=(a1+(a2+a3),b1+(b2+b3)+a1(a2a3))
=(a1,b1)+[(a2,b2)+(a3,b3)]
3.(a,b)+(0,0)=(a,b)
4.(a,b)+(﹣a,a^2－b)=(0,0)
5.1•(a,b)=(a,b)
6.k[l(a,b)]=k[la,lb+l(l-1)a^2/2]
=[(kl)a,klb+kl(l-1)a^2/2+k(k-1)l^2a^2/2]
={(kl)a,(kl)b+(kl)[(kl)-1]a^2/2}
=(kl)•(a,b)
7.(k+l)•(a,b)={(k+l)a,(k+l)b+(k+l)[(k+l)-1]a^2/2}
={ka+la,[kb+k(k-1)a^2/2]+[lb+l(l-1)a^2/2]+(ka)(la)}
=[ka,kb+k(k-1)a^2/2]+[la,lb+l(l-1)a^2/2]
=k(a,b)+l(a,b)
8.k[(a1,b1)+(a2,b2)]=k(a1+a2,b1+b2+a1a2)
=[ka1+ka2,kb1+kb2+ka1a2+k(k-1)(a1+a2)^2/2]
={ka1+ka2,[kb1+k(k-1)a1^2/2]+[kb2+k(k-1)a2^2/2]+k^2a1a2}
=[ka1,kb1+k(k-1)a1^2/2]+[ka2,kb2+k(k-1)a2^2/2]
=k(a1,b1)+k(a2,b2)
且(a1,b1)+(a2,b2)与k(a,b)均为二元有序实数组,即该集合关于该加法和数乘封闭,所以这个集合是一个线性空间.