1. ## Question about inner products

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 ${\left\langle {x,y} \right\rangle _1}$ and ${\left\langle {x,y} \right\rangle _2}$ are inner products defined on V.

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

I've tried methods like ${\left\langle {x,y} \right\rangle _1} = {\left\langle {x,y + x - x} \right\rangle _1} = {\left\langle {x,y} \right\rangle _1} + {\left\langle {x,x} \right\rangle _1} - {\left\langle {x,x} \right\rangle _1}$, 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 ${\left\langle {x,y} \right\rangle _1}$ and ${\left\langle {x,y} \right\rangle _2}$ are inner products defined on V.

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

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

Thanks!

Develop $_1=_2$...as simple as that.

Tonio