I'm having trouble with this for some reason. If [tex]A:\mathcal{H}\to \mathcal{H}[/tex] is a bounded operator between Hilbert spaces, the norm of [tex]A[/tex] is(adsbygoogle = window.adsbygoogle || []).push({});

[tex] ||A|| = \inf\limits_{\psi \neq 0} \frac{||A\psi||}{||\psi||}[/tex].

My trouble is in verifying that [tex]||A||[/tex] is in fact a bound for [tex]A[/tex] in the sense that [tex]||A\psi|| \leq ||A|| ||\psi||[/tex]. I'm actually not even sure if that's true, but I was able to verify this by the definition given here http://en.wikipedia.org/wiki/Operator_norm. I basically just want to make sure the definitions are equivalent. The trouble is that if [tex]\psi\in \mathcal{H}[/tex], then by definition [tex]||A|| \leq \frac{||A\psi||}{||\psi||}[/tex] and this gives the incorrect inequality.

Did I overlook something?

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# Norm of a Bounded Operator

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