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Math Help - Function Spaces

  1. #1
    Senior Member bugatti79's Avatar
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    Function Spaces

    Folks,

    a) Does the function space || ||:C[0,1] \rightarrow R defined by ||f||=|f(1)-f(0)| define a norm on C[0,1]

    b)If it does, prove all the axioms for a norm hold. If not, demonstrate by an example some axiom which fails

    How do I start a)....I have no idea....?

    THanks
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  2. #2
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    Re: Function Spaces

    let  f(x) = x^2 - x. then ||f|| = 0, but f ≠ 0.
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  3. #3
    Senior Member bugatti79's Avatar
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    Re: Function Spaces

    Quote Originally Posted by Deveno View Post
    let  f(x) = x^2 - x. then ||f|| = 0, but f ≠ 0.
    What I have in my notes is that a norm on a vector space V is a function || ||:V to R satisfying x and y in V and for all scalars alpha. One of the axioms states that ||x||=0 IFF x vector = zero vector.

    Hence the above function does not define a norm...not sure how to rigorously prove it though...?

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  4. #4
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    Re: Function Spaces

    Quote Originally Posted by bugatti79 View Post
    What I have in my notes is that a norm on a vector space V is a function || ||:V to R satisfying x and y in V and for all scalars alpha. One of the axioms states that ||x||=0 IFF x vector = zero vector.

    Hence the above function does not define a norm...not sure how to rigorously prove it though...?

    THanks
    To show something is false you only need to provide one counter example as Deveno did.
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  5. #5
    Senior Member bugatti79's Avatar
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    Re: Function Spaces

    Quote Originally Posted by TheEmptySet View Post
    To show something is false you only need to provide one counter example as Deveno did.
    So thats the answer! Ok, just one other question...where is the rule that that a function cannot be = 0?

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    Re: Function Spaces

    Quote Originally Posted by bugatti79 View Post
    So thats the answer! Ok, just one other question...where is the rule that that a function cannot be = 0?

    Thanks
    A function can equal zero, but one of the axioms of a Norm is that the only objected that can have norm 0 is the zero of the space.

    The function f(x)=0 is the zero of C[0,1] Deveno's example gives another function that is not 0 that has lenth zero.
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  7. #7
    Senior Member bugatti79's Avatar
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    Re: Function Spaces

    Quote Originally Posted by TheEmptySet View Post
    A function can equal zero, but one of the axioms of a Norm is that the only objected that can have norm 0 is the zero of the space.

    The function f(x)=0 is the zero of C[0,1] Deveno's example gives another function that is not 0 that has lenth zero.
    Ok, just to make sure I understand correctly. The given function does not define a norm on C[0,1] because the function itself must be 'zero' Ie, f is 0...?
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  8. #8
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    Re: Function Spaces

    Quote Originally Posted by bugatti79 View Post
    Ok, just to make sure I understand correctly. The given function does not define a norm on C[0,1] because the function itself must be 'zero' Ie, f is 0...?
    Yes one of the properties of a norm is ||f||=0 iff f is the zero of the space. Otherwise you get a psudonorm if that is the only property that fails.
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