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Math Help - does this define a norm on R[0,1]?

  1. #1
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    does this define a norm on R[0,1]?

    R[0,1] is the vector space of all integrable functions on [0,1]

     \|f\|_{1}=\int_{0}^{1}{|f|}

    does this define a norm?

    i think yes. because i cant find any axiom fails
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi,

    How did you show that \|f\|_1=0 implies f=0 ?
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  3. #3
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    ob

    isnt it obvious?

    u just show the upper riemann sum=lower riemann sum=0 ( by integrale)

    therefore, sup f=inf f=0
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by szpengchao View Post
    u just show the upper riemann sum=lower riemann sum=0 ( by integrale)

    therefore, sup f=inf f=0
    This doesn't work for <br />
f:x \mapsto \begin{cases} 0 &\mathrm{if}\,\,0\leq x<1 \\ 1& \mathrm{if}\,\, x=1\\ \end{cases} since one has f\neq 0 but \int_0^1|f(x)|\,\mathrm{d}x=0.
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