Compute the norm of $\phi$

Asked Mar 29 '19 at 17:17

Active Mar 29 '19 at 17:17

Viewed 54 times

We have , $E=C([0,1],\mathbb{R})$ equipped with norm $\left \| . \right \|_\infty $ and $\phi:E\rightarrow E$

$\phi(f)(x)=\int_{0}^{x}tf(t)dt+x\int_{x}^{1}f(t)dt$, for any $x\in [0,1]$

I proved that $\phi$ is linear and continuous (Lipschitz ) (choose k=1)

Using fundamental relation, i got $\left \| \phi \right \|\leq 1$

Now I know that I should find a function such that $\left \| \phi \right \|\geq 1$, how can I find it ?

asked Mar 29 '19 at 17:17

Pedro Alvarès

Hint: You can similarly prove $|\phi|\le\frac12$. – Berci Mar 29 '19 at 17:33
If I take f(t) constant right ? – Pedro Alvarès Mar 29 '19 at 17:57
I mean for all $f$. You can sharpen your estimation. I guess the norm will be $\frac12$. – Berci Mar 29 '19 at 18:01
You mean if k=0.5 (when proving Lipschitz) , I get your result ? – Pedro Alvarès Mar 29 '19 at 18:17

0 Answers0