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Homework Help: Equivalent norms?

  1. Oct 17, 2008 #1
    Suppose that ||f||= int 01| f(x) | dx and f is a piecewise continuous linear function on the interval [0,1]. If ||| f ||| = int 01 x | f(x) | dx, determine if the two norms are equivalent.

    I know the first defines a norm, and the space is not complete. Can anyone offer any hints as to solving this problem?
     
  2. jcsd
  3. Oct 17, 2008 #2
    Well, first of, when are two norms defined to be equivalent? If you know that, one part of the definition is very easily verified. For the other part consider the function

    [tex]
    f_{\varepsilon}(x)=x\chi_{[0,\varepsilon]}(x)
    [/tex]
    What is then [itex]\frac{||f_{\varepsilon}||}{|||f_{\varepsilon}|||}[/itex]? Conclusion?
     
    Last edited: Oct 17, 2008
  4. Oct 17, 2008 #3
    i don't quite follow... is that function in the space?
     
  5. Oct 17, 2008 #4

    morphism

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    [itex]\chi_{[0,\varepsilon]}[/itex] is the characteristic function of [itex][0,\varepsilon][/itex], i.e.

    [tex]\chi_{[0,\varepsilon]}(x) = \begin{cases} 1 & \text{ if } x \in [0,\varepsilon] \\ 0 & \text{ if } x \in (\varepsilon, 1] \end{cases}.[/tex]
     
  6. Oct 17, 2008 #5
    That function is certainly piecewise continous and (piecewise) linear und thus in the space you mentioned.
    Thanks for supplying this additional, crucial piece of information that by [itex]\chi_A[/itex] I meant the indicator function of A.
     
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