Hilbert, Inner Product
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Mar30-12, 02:02 PM
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
This isn't Axiom 1. It has to do with <x, x>.
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
Axiom 3 <y,x>= complex of <x,y>
Don't you mean "conjugate transpose"?
Since the underlying vector space is R
, all you need to show is that <x, y> = <y, x>.
<y,x>=(y1x1+y2x2+y3x3) but y complex =y and x complex=x in R therefore
= (y1x1 complex+y2x2 complex +x3y3 complex)
Axiom 4 <x+y,z>=<x,z>+<y,z>, let z=(z1,z2,z3) in R^3