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:

    “Every notable advance in technique or organization has to be paid for, and in most cases the debit is more or less equivalent to the credit. Except of course when it’s more than equivalent, as it has been with universal education, for example, or wireless, or these damned aeroplanes. In which case, of course, your progress is a step backwards and downwards.”
    —Aldous Huxley (1894–1963)

    “What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
    —G.C. (Georg Christoph)