Proving the norm supremum formula in a Hilbert space

[Kindayr]

Homework Statement

Let $H$ be a Hilbert space. Prove $\Vert x \Vert = \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert}$

The Attempt at a Solution

First suppose $x = 0$. Then we have $\sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert (0,y) \vert}{\Vert y \Vert} = \sup_{0\neq y\in H}\frac{\vert 0 \vert}{\Vert y \Vert} = 0 = \Vert 0 \Vert$.

Now suppose $x \neq 0$. Then $\Vert x \Vert = \sqrt{(x,x)} = \frac{\sqrt{(x,x)} \cdot \sqrt{(x,x)}}{\sqrt{(x,x)}} = \frac{\vert (x,x)\vert}{\Vert x \Vert} \leq \sup_{0\neq y\in H}\frac{\vert (x,y) \vert}{\Vert y \Vert}$.

Now I just can't do the reverse inequality. Any help is much appreciated.

 

[jbunniii]

Cauchy-Schwarz?

 

[Kindayr]

Cauchy-Schwarz?

omg how do i call myself a math major.

thank you.