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}

Read more about this topic:  Operator Norm

Famous quotes containing the words equivalent and/or definitions:

    “The notion that one will not survive a particular catastrophe is, in general terms, a comfort since it is equivalent to abolishing the catastrophe.”
    —Iris Murdoch (b. 1919)

    “The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
    —Pamela Patrick Novotny (20th century)