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Cartesian Product of Metric Spaces

  1. Sep 6, 2006 #1


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    Hello everyone.

    I read in a book that for metric spaces [itex](X, \rho), (Y, \sigma)[/itex] we can form the metric space [itex] (X \times Y, \tau_p) [/itex], for [itex]1 \leq p < \infty[/itex] where [itex]\tau_p[/itex] is given by:

    [tex]\tau_p((x_1,y_1), (x_2,y_2)) = (\rho(x_1,x_2)^p + \sigma(y_1,y_2)^p)^\frac{1}{p}[/tex]

    I can easily verify the positivity and symmetry of [itex]\tau_p[/itex] but verifying the triangle innequality is a bit tricky. Any suggestions? Here is what i've tried with no luck:

    [tex](\rho(x_1,x_2)^p + \sigma(y_1,y_2)^p)^\frac{1}{p} \leq \rho(x_1,x_2) + \sigma(y_1,y_2) [/tex]

    since rho and sigma are metrics they each satisfy the triangle innequality in their respective spaces and so we have for any (x3,y3) in X*Y,

    RHS [tex] \leq \rho(x_1,x_3) + \rho (x_3,x_2) + \sigma(y_1,y_3) + \sigma(y_3,y_2) [/tex]

    and I'm stuck. Thanks in advance for your help.
    Last edited: Sep 6, 2006
  2. jcsd
  3. Sep 6, 2006 #2


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    How did you prove the Euclidean metric on R² satisfies the triangle equality? Wouldn't the same proof work here?
  4. Sep 7, 2006 #3


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    Thanks Hurkyl. It was even easier than that since I had already proved the Minkowski inequality, so I just had to recognize that I could apply that here. However, It was your comment that led me to recognize this. Thank You!
    Last edited: Sep 7, 2006
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