# Thread: Norm of powers of the Volterra Operator

1. ## Norm of powers of the Volterra Operator

Hey there,

For a course in Functional Analysis we were asked to show that the norm of the n-th power of the Volterra operator is less than or equal to 1 over n!.

$\displaystyle$V: C[0,1] \rightarrow C[0,1]$\\(Vf)(s) = \displaystyle{\int_0^s f(t)dt}$
Show $\displaystyle$\left|\left|V^n\right|\right| \leq \frac{1}{n!}$$Now I know how the operator norm is defined \displaystyle \left|\left|T\right|\right| = \displaystyle{\sup_{\left|\left|x\right|\right|=1} } \left|\left|Tx\right|\right| and I've been throwing it at this problem for at least 3 hours, but I can't seem to crack it. I've so far assumed that induction would be the path to choose but I'm starting to doubt that, yet I see no other method. I hope someone is able to nudge me in the right direction. Thanks, Insensus 2. ## Re: Norm of powers of the Volterra Operator Originally Posted by Insensus Hey there, For a course in Functional Analysis we were asked to show that the norm of the n-th power of the Volterra operator is less than or equal to 1 over n!. \displaystyle V: C[0,1] \rightarrow C[0,1]\\(Vf)(s) = \displaystyle{\int_0^s f(t)dt} Show \displaystyle \left|\left|V^n\right|\right| \leq \frac{1}{n!}$$

Now I know how the operator norm is defined

4. ## Re: Norm of powers of the Volterra Operator

That's very nice, thanks.