Hi, I have a question I need to do that look quite axiom-based, but I've been working through it for a while and I just can't seem to get out the answer that I need!

We assume that V is a real vector space, and $\displaystyle {\left\langle {x,y} \right\rangle _1}$ and $\displaystyle {\left\langle {x,y} \right\rangle _2}$ are inner products defined on V.

Assuming that $\displaystyle \forall x \in V,{\rm{ }}{\left\langle {x,x} \right\rangle _1} = {\left\langle {x,x} \right\rangle _2}$, I want to show that $\displaystyle \forall x,y \in V,{\rm{ }}{\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y} \right\rangle _2}$.

I've tried methods like $\displaystyle {\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y + x - x} \right\rangle _1} = {\left\langle {x,y} \right\rangle _1} + {\left\langle {x,x} \right\rangle _1} - {\left\langle {x,x} \right\rangle _1}$, but this doesn't seem to lead me anywhere. Can anyone help?

Thanks!

2. Originally Posted by lalala23
Hi, I have a question I need to do that look quite axiom-based, but I've been working through it for a while and I just can't seem to get out the answer that I need!

We assume that V is a real vector space, and $\displaystyle {\left\langle {x,y} \right\rangle _1}$ and $\displaystyle {\left\langle {x,y} \right\rangle _2}$ are inner products defined on V.

Assuming that $\displaystyle \forall x \in V,{\rm{ }}{\left\langle {x,x} \right\rangle _1} = {\left\langle {x,x} \right\rangle _2}$, I want to show that $\displaystyle \forall x,y \in V,{\rm{ }}{\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y} \right\rangle _2}$.

I've tried methods like $\displaystyle {\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y + x - x} \right\rangle _1} = {\left\langle {x,y} \right\rangle _1} + {\left\langle {x,x} \right\rangle _1} - {\left\langle {x,x} \right\rangle _1}$, but this doesn't seem to lead me anywhere. Can anyone help?

Thanks!

Develop $\displaystyle <x-y,x-y>_1=<x-y,x-y>_2$...as simple as that.

Tonio