MHB Bounded linear functional question? Real Analysis

[Juliayaho]

Consider the functional Tf = f(5) - i f(7). If we take the domain T to be C_0(ℝ) with supremum norm, is T a bounded linear functional?
What if we take the domain to be C_c(ℝ) with L^2 norm || . ||_2?I know I should post what I have so far but this time I have no idea because I had to missed 2 class sections... So I need to do some practice problems but I'm lost :S Any help would be appreciated thanks!

 

[Opalg]

Consider the functional Tf = f(5) - i f(7). If we take the domain T to be C_0(ℝ) with supremum norm, is T a bounded linear functional?
What if we take the domain to be C_c(ℝ) with L^2 norm || . ||_2?I know I should post what I have so far but this time I have no idea because I had to missed 2 class sections... So I need to do some practice problems but I'm lost :S Any help would be appreciated thanks!

If you missed some classes, the first thing you need to do is to make sure that you understand what this problem is about. So, given a function $f$ in the appropriate space, what is the definition of the supremum norm of $f$, and what is the definition of $\|f\|_2$? Next, what is a bounded linear functional? Write down the definition of what it means for $T$ to be a bounded linear functional, for each of those two norms.

Once you know what the definitions are, try to see whether the functional $T(f) = f(a)$ (for some fixed number $a$) is a bounded linear functional, for each of the two norms. You may find that general problem a bit easier than dealing with the particular cases when $a=5$ and $a=7$.

Come back here if you are still having difficulties once you have sorted out what the definitions are.

 

[Poirot1]

Consider the functional Tf = f(5) - i f(7). If we take the domain T to be C_0(ℝ) with supremum norm, is T a bounded linear functional?
What if we take the domain to be C_c(ℝ) with L^2 norm || . ||_2?I know I should post what I have so far but this time I have no idea because I had to missed 2 class sections... So I need to do some practice problems but I'm lost :S Any help would be appreciated thanks!

T:E->C is a bounded linear functional if T is linear i.e. T(ax+by)=aT(x)+bT(y) for all a,b in C and x,y in E, and T is bounded i.e there exists k>0 such that ||T(x)||<_k||x|| for all x in E. The norm of T is ||T||=inf{k>0:||T(x)||<_k||x|| }. If E={0}, then ||T||=0 is easy to see. Otherwise there exist vectors in E with norm 1, and it can be shown that

||T||=Sup{||Tx||:||x||=1}. Also it can be shown that ||Tx||<_||T||||x|| for all x in E.

Hopefully these notes will help.

 

[Juliayaho]

Thank you all for your help :)