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Math Help - operator norms for linear functionals

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by vinnie100 View Post
    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].
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  3. #3
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    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.
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  4. #4
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    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\|.
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  5. #5
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    Thanks very much for your answer! This makes perfect sense!
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