Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if $\lVert x+\alpha y\rVert \geq\lVert x\rVert$ for all scalar $\alpha$?
I have proved that $\langle x,y\rangle=0$ implies $\lVert x+\alpha y\rVert\geq \lVert x\rVert$, but don't know how to prove the converse. I just try to show that $\lVert x+\alpha y\rVert\geq \lVert x\rVert$ implies $\lVert x+\alpha y\rVert=\lVert x-\alpha y\rVert$ (but still not success) since this equality equivalent with $\langle x,y\rangle=0$.
Is there any solution of this problem using this information: ∥x+αy∥=∥x−αy∥ if and only if =0 ? Thanks.