\|A\|=\sup \{ \|A(x,y)\|_{\infty} \mid x^2+y^2=1 \} = \sup \{ \|A(\cos \varphi, \sin \varphi )\|_{\infty} \mid \varphi \in \mathbb{R} \} = \sup \{ \max \{ | \cos \varphi + \sin \varphi | , | \cos \varphi - \sin \varphi | \} \mid \varphi \in \mathbb{R} \} = \sup \{ | \cos \varphi + \sin \varphi | \mid \varphi \in \mathbb{R} \} = \sup \{ \sqrt{2} | \frac{\sqrt{2}}{2} \cos \varphi + \frac{\sqrt{2}}{2} \sin \varphi | \mid \varphi \in \mathbb{R} \} = \sup \{ \sqrt{2} | \cos (\varphi - \frac{\pi}{4}) | \mid \varphi \in \mathbb{R} \} = \sqrt{2}