After justifying the existence of the suprema, show that: $$\sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| } = \sup _ { \substack { x \in E \\ \| x \| = 1 } } \| u ( x ) \| .$$
After justifying the existence of the suprema, show that:
$$\sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| } = \sup _ { \substack { x \in E \\ \| x \| = 1 } } \| u ( x ) \| .$$