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Math Help - Metric of a product

  1. #1
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    Metric of a product

    Suppose that (X, \rho ),(Y, \gamma ) are metric spaces, define  ( X \times Y , \phi ) with  \phi ((x_1,y_1),(x_2,y_2)) = \sqrt { ( \rho (x_1, x_2))^2+ ( \gamma (y_1,y_2))^2 }

    Show that  \phi ((x_1,y_1),(x_2,y_2)) + \phi ((x_2,y_2),(x_3,y_3)) \geq \phi ((x_1,y_1),(x_3,y_3))

    Proof so far.

    I know that  \rho (x_1,x_2) + \rho (x_2,x_3) \geq \rho (x_1,x_3)
    and  \gamma (y_1,y_2) + \gamma (y_2,y_3) \geq \gamma (y_2,y_3)

    And I'm trying to show that  \sqrt { ( \rho (x_1, x_2))^2+ ( \gamma (y_1,y_2))^2 } + \sqrt { ( \rho (x_2, x_3))^2+ ( \gamma (y_2,y_3))^2 }  \geq \sqrt { ( \rho (x_1, x_3))^2+ ( \gamma (y_1,y_3))^2 }

    How should I proceed? Thank you!
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello,
    Quote Originally Posted by tttcomrader View Post
    How should I proceed?
    We have
     \sqrt {\rho (x_1, x_3)^2+\gamma (y_1,y_3)^2 }  \leq \sqrt { \rho (x_1, x_2)^2+ \rho (x_2, x_3)^2 + \gamma (y_1,y_2)^2+ \gamma (y_2,y_3)^2 }.
    Now use this result :
    If u and v are non-negative real numbers then \sqrt{u+v}\leqslant\sqrt{u}+\sqrt{v}
    and you are done.
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