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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Norm of a Bounded Operator

**Physics Forums | Science Articles, Homework Help, Discussion**