Thread: 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!

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].

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.

1. Let the linear functional A: C[-1,1]-> R be defined by
for all f in C[-1,1].

Find the operator norm of A.

We want to find , which (putting it loosely, in words) is the biggest value that can take, for f in the unit ball of C[–1,1]. So we want to maximise subject to the condition that for all t in [–1,1].

Notice that , 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
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 is positive throughout the interval [–1,1]. But in problem 2., the function changes sign in the two halves of the interval. To maximise , 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 .

Thanks very much for your answer! This makes perfect sense!