Operator Norm - Equivalent Definitions

Equivalent Definitions

One can show that the following definitions are all equivalent:

\begin{align} \|A\|_{op} &= \inf\{c \ge 0 : \|Av\| \le c\|v\| \mbox{ for all } v\in V\} \\ &= \sup\{\|Av\| : v\in V \mbox{ with }\|v\| \le 1\} \\ &= \sup\{\|Av\| : v\in V \mbox{ with }\|v\| = 1\} \\ &= \sup\left\{\frac{\|Av\|}{\|v\|} : v\in V \mbox{ with }v\ne 0\right\}. \end{align}

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