Prove $||A||_2 = max_{x \neq 0, y \neq 0}\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}\}$ for $A\in \mathbb{K}^{n\times m}$

Question

I'd like to prove that the spectral norm of a matrix that is not necessarily square can be written as the following subordinate norm

$||A||_2 = max\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}, y \in \mathbb{K}^n\backslash\{0\}, x \in \mathbb{K}^m\backslash\{0\} \}$ for $A\in \mathbb{K}^{n\times m}$

where $\mathbb{K}$ is $ \mathbb{R}$ or $\mathbb{C}$

I started doing the following: $||A||_2 = max_{||v||_2 = 1} ||Av||_2 = max_{||v||_2 = 1}\langle z,Av\rangle^2$ for $z=Av$ but I feel like I'm going down the wrong path. Any help would be much appreciated!

Answer 1

HINT: $$ \|A\|_2=\max_{\|v\|_2=1}\|Av\|_2, \quad \|w\|_2=\max_{\|z\|_2=1}\left|\langle z,w\rangle\right|. $$ Use the second in the first.