Show an inequality

Question

Show that on the Hilbert space: $$x\bot y $$ if and only if $$ \Vert x + Cy\Vert \ge \Vert x \Vert,$$ $\forall C\in \mathbb R.$

Answer 1

Observe that, for given $C\in \mathbb{R},$ $$\|x+Cy\|^2=\left<x+Cy,x+Cy\right>=\|x\|^2+C^2\|y\|^2\geq \|x\|^2 $$ due to the orthogonality ($x\perp y \implies \left<x,y \right>=0.$)

Take the square root, then you get $$\|x+Cy\|\geq \|x\|$$

Answer 2

$||x+Cy||^{2}=||x||^{2}+2C\langle x,y\rangle+C^{2}||y||^{2}$

$||x+Cy||^{2}\ge ||x||^{2} \iff 2C\langle x,y\rangle+C^{2}||y||^{2}\ge 0$

This associated equation is an quation of second degree has two real roots $C_{1}=0$, $C_{2}=\frac{2\langle x,y\rangle}{||y||^{2}}$. So the inequality is always true for all $C\in\mathbb{R} \iff C_{1}=C_{2} \iff x\perp y$