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Math Help - Metric of C[0,1]

  1. #1
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    Metric of C[0,1]

    Consider the set of C[0,1], the set of all continuous functions on [0,1], and define p(f,g)= \int ^1 _0 \mid f-g \mid dx .

    Prove that if p(f,g)=0, then f = g.


    If p(f,g)=0, then I have  \int ^1 _0 \mid f(x)-g(x) \mid dx =0.

    But now, what can I do with the absolute values? Thanks!
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by tttcomrader View Post
    But now, what can I do with the absolute values?
    Use this property: If h\in C[0,1] is a non-negative function such that \int_0^1h(x)\,\mathrm{d}x = 0 then h=0.
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