$(X,\langle.,.\rangle)$ is an inner product space over $\mathbb{R}$. If $x, y \in X$ where the induced norms of $x$ and $y$ are equal, prove that $x+y$ and $x-y$ are orthogonal.
So I want to show that $\langle x+y,x-y\rangle=0$.
I get that: $$\langle x+y,x-y\rangle=\langle x,x \rangle - \langle y,y \rangle.$$ But I don't know how to show this is equal to $0$. I feel like I'm missing something important regarding inner products...
Since $X$ is a real inner-product space, there is not conjugation in the inner products, so we have: $$\langle x+y,x-y\rangle=\langle x,x\rangle +\langle x,y\rangle -\langle x,y\rangle - \langle y,y\rangle=\langle x,x\rangle -\langle y,y\rangle=||x||^2-||y||^2=0$$ Since $||x||=||y||$ as stated in the question.