Thread: Different Forms of Operator Norm

1. Different Forms of Operator Norm

I was given the very first inf definition, and have to prove that 3rd is the same as the given one. What I tried is taking sup on both sides of llAvll =< llAll llvll, but failed. Any help would be greatly appreciated.
Thanks

2. Let $\|A\|_0=\sup\nolimits_{\|v\|=1}\|Av\|$ and let $\|A\|_1=\inf\{c:\|Av\|\leq c\|v\|\textrm{ for all }v\in V\}$.

Consider any $c$ such that $\|Av\|\leq c\|v\|$ for all $v\in V$ and choose any $u\in V$ with $\|u\|=1$.

Then $\|Au\|\leq c\|u\|=c$ and so $\sup\nolimits_{\|u\|=1}\|Au\|\leq c$, i.e. $\|A\|_0\leq c$. Taking the inf over all such $c$, we deduce that $\|A\|_0\leq \|A\|_1$.

If $v\in V$ is non-zero, let $u=v/\|v\|$. Then $\|u\|=1$ and so $\|Au\|\leq\|A\|_0$.

It follows by linearity that $\|Av\|\leq\|A\|_0\|v\|$ for all non-zero $v\in V$ and therefore for all $v\in V$, since it is clearly true when $v=0$.

Hence $\|Av\|\leq c\|v\|$ for all $v\in V$ when $c=\|A\|_0$. By the definition of $\|A\|_1$, we see that $\|A\|_1\leq\|A\|_0$.

Thus we have $\|A\|_0=\|A\|_1$.

3. Thank you very much