# Thread: operator norms for linear functionals

1. ## operator norms for linear functionals

I have started studing linear functionals and have looked at shift operators and multiplication operatotrs briefly. However, I have come across a problem as to how to answer the following questions:

1. Let the linear functional A: C[-1,1]-> R be defined by
A(f) = integral [t^2. f(t)] dt for all f in C[0,1] [limits of integral are +1,-1]

Find the operator norm of A.

2. Let the linear functional A: C[-1,1]-> R be defined by
A(f) = integral [t f(t)] dt for all f in C[0,1] [limits of integral are +1,-1]

Find the operator norm of A.

If someone could show a method of as how to do Qu1, I think I could apply that idea to do Qu2. Sorry for the trouble but would be very grateful if you could help

Thanks!

2. Originally Posted by vinnie100
I have started studing linear functionals and have looked at shift operators and multiplication operatotrs briefly. However, I have come across a problem as to how to answer the following questions:

1. Let the linear functional A: C[-1,1]-> R be defined by
A(f) = integral [t^2. f(t)] dt for all f in C[0,1] [limits of integral are +1,-1]

Find the operator norm of A.
You need to clarify this problem a bit.

First, what is the domain space of the operator A? Is it C[–1,1] or is it C[0,1]? (It looks to me as though C[0,1] is a typo and that you meant to write C[–1,1] both times.)

Second, assuming that the domain space is C[–1,1], you have not specified the norm on that space. There are many different norms on the space C[–1,1], and the norm of the functional A will depend on the norm used in C[–1,1].

3. Oh I am really sorrry.
Yes the first one is a typo - the domain is all C[-1,1] of continuous functions.
We take continuous functions to be equipped with the sup norm.
i.e. !f! = sup |f(t)| for a<=t<=b

[in fact I never knew that this was a rule in our course and that other norms could be taken!]

Hope this helps
Thanks again.

4. 1. Let the linear functional A: C[-1,1]-> R be defined by
$A(f) = \int_{-1}^1t^2f(t)\,dt$ for all f in C[-1,1].

Find the operator norm of A.
We want to find $\|A\|$, which (putting it loosely, in words) is the biggest value that $\|A(f)\|$ can take, for f in the unit ball of C[–1,1]. So we want to maximise $\Bigl|\int_{-1}^1t^2f(t)\,dt\Bigr|$ subject to the condition that $|f(t)|\leqslant1$ for all t in [–1,1].

Notice that $\Bigl|\int_{-1}^1t^2f(t)\,dt\Bigr|\leqslant \int_{-1}^1t^2\,dt = \frac23$, and that this maximum value 2/3 is attained for the function f(t)=1 (for all t).

2. Let the linear functional A: C[-1,1]-> R be defined by
$A(f) = \int_{-1}^1tf(t)\,dt$ for all f in C[-1,1].

Find the operator norm of A.
I'm not going to do this one for you, except to point out that it is more complicated. In problem 1., the function $t^2$ is positive throughout the interval [–1,1]. But in problem 2., the function $t$ changes sign in the two halves of the interval. To maximise $\Bigl|\int_{-1}^1tf(t)\,dt\Bigr|$, you would ideally like to take the function f(t) to be –1 in the interval [–1,0], and +1 in the interval [0,1]. But then f(t) would not be continuous at t=0. So you have to approximate that discontinuous function by continuous functions in order to find $\|A\|$.