I have just realized that I accidently put it in wrong sub forum. This should be in 'calculus and beyond'. 1. The problem statement, all variables and given/known data Prove the function <x,y>=x_1y_1+x_2y_2+x_3y_3 defines an inner product space on the real vector space R^3 where x=(x1,x2,x3) and y=(y1,y2,y3) 3. The attempt at a solution Axiom 1 <x,y> >=0 since we have that x_n and y_n for n=1,2,3 are in R Axiom 2a <x,y> =x1y1+x2y2+x3y3, then <x,y>=0 iff x_n and y_n for n=1,2,3 both =0 Axiom 2b <ax,y>=a<x,y> <ax,y> = ax1y1+ax2y2+ax3y3 = a(x1y1+x2y2+x3y3) =a<x,y> Axiom 3 <y,x>= complex of <x,y> <y,x>=(y1x1+y2x2+y3x3) but y complex =y and x complex=x in R therefore = (y1x1 complex+y2x2 complex +x3y3 complex) =<y,x> complex =<x,y> complex Axiom 4 <x+y,z>=<x,z>+<y,z>, let z=(z1,z2,z3) in R^3 <x+y,z>=(x1+y1+x2+y2+x3+y3)(z1+z2+z3) =x1z1 +x2z2+x3z3+y1z1+y2zy3z3 =<x,z>+<y,z> ...?