Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the Banach space $B=C[-1,1]$, with $\sup$ norm, for $f\in B$ define $\tilde f(x)=f(|x|)$, $T:B\rightarrow B, T(f)=\tilde f$ we need to show $T$ is a bounded linear operator on $B$, what is $||T||?$

$T(cf+g)= (f+g)(|x|)=cf(|x|)+ g(|x|)=c\tilde f+\tilde g$ so $T$ is linear as we know continous functions over compact set is bounded so clearly $T$ is bounded?

I am not able to determine $||T||$.

share|improve this question
    
The domain of $T$ is $B$ not $[-1,1]$. –  Rudy the Reindeer Nov 2 '12 at 8:31

1 Answer 1

up vote 3 down vote accepted

$$ \|T\| = \sup_{\|f\|_\infty = 1, f \in B} \|Tf\|_\infty = \sup_{\|f\|_\infty = 1, f \in C[0,1]} \|f\|_\infty = 1$$

Hence $T$ is bounded.

share|improve this answer
    
just to recall myself, $||T||=sup_{||x||=1}||T(x)||$? –  El Angel Exterminador Nov 2 '12 at 8:40
    
@Flute Well, that is one of many. –  Rudy the Reindeer Nov 2 '12 at 8:42

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.