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Math Help - metric space

  1. #1
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    metric space

    Hi can someone please help me with the following question:

    metric space-untitled123.jpg
    i know and understand the definition of a metric space:
    metric space-untitled.jpg

    I can prove easily M1 and M2 but M3 is giving me problems. Can someone please help me.

    M1: e(p,p)=0 by defintion.
    e(p,q)>0 if p does not equal to q

    M2: e(p,q)= (lpl+lql)=(lql+lpl)=e(q,p)

    M3: really need help

    Thanks

    nerdo
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  2. #2
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    Is this what you mean?

    [MATh]
    \varsigma (x,y) \leqslant \varsigma (x,z) + \varsigma (z,y) [/tex]
    Last edited by Plato; October 7th 2010 at 12:15 PM.
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  3. #3
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    yes i am finding it difficult to prove this. Can you please help me.
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  4. #4
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    Can you explain where you are having trouble?
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  5. #5
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    i don't understand how the case if x=y but not z still holds
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  6. #6
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    Is this true: \varsigma (x,z) = \left\| x \right\|_2  + \left\| z \right\|_2 \;\& \,\varsigma (z,y) = \left\| z \right\|_2  + \left\| y \right\|_2 ?

    What if you were to add those together?

    Is 2\left\| z \right\|_2  \geqslant 0?
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  7. #7
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    Thanks for the help. It made my life much easier.
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  8. #8
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    Quote Originally Posted by nerdo View Post
    Thanks for the help. It made my life much easier.
    Remarkable! Hey, Plato, can you make my life much easier?
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