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 :)

 

[blue_raver22]

I am not an expert in Real Analysis, but I can provide some insight on bounded linear functionals based on my understanding of mathematical concepts.

A bounded linear functional is a linear map from a normed vector space to its underlying field that satisfies certain properties. In this case, the functional Tf = f(5) - i f(7) is a linear map from the space of continuous functions C_0(ℝ) to the complex numbers. The supremum norm ||f||_∞ is the maximum value of a function over its domain, and in this case, it is used to measure the size of the functions in C_0(ℝ).

To determine if T is a bounded linear functional, we need to check if it satisfies two properties: linearity and boundedness. Linearity means that for any two functions f and g in C_0(ℝ) and any scalar a, T(af+g) = aTf + Tg. In this case, we can easily see that T satisfies linearity since T(af+g) = (af+g)(5) - i (af+g)(7) = a(f(5)-if(7)) + (g(5)-ig(7)) = aTf + Tg.

The second property, boundedness, means that there exists a constant M such that ||Tf|| ≤ M||f|| for all f in C_0(ℝ). To determine if T is bounded, we need to find the supremum norm of Tf and compare it to the supremum norm of f. However, since the supremum norm of Tf involves evaluating the function at specific points (5 and 7), it is not clear how to find M. Therefore, it is not possible to determine if T is a bounded linear functional on C_0(ℝ) with the given information.

Now, let's consider the domain to be C_c(ℝ) with L^2 norm || . ||_2. In this case, the L^2 norm measures the size of a function by considering its integral over the entire domain. To check if T is a bounded linear functional on C_c(ℝ), we can use the Cauchy-Schwarz inequality, which states that for any two functions f and g in C_c(ℝ), |f(5)g(5) + f(