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I saw somewhere that Cartesian product $X = X_1 \times X_2$ of two metric spaces $(X_1,d_1)$ and $(X_2,d_2)$ can be made into a metric space $(X,d)$ like following:

$d(x,y) = (d_1(x_1,y_1)^p + d_2(x_2,y_2)^p)^{1/p}$

How do I prove $(X,d)$ is metric space? More specifically, how to prove triangular inequality?

[Note: $p\ge1$, $x=(x_1,x_2)$ and $y= (y_1,y_2)$]

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  • $\begingroup$ the latter part $d_1 d_2$ is it $d_2(x_2, y_2)$? $\endgroup$ – Santosh Linkha Sep 27 '13 at 14:16
  • $\begingroup$ yes sir,i corrected it $\endgroup$ – Raja Sekar Sep 27 '13 at 14:20
  • $\begingroup$ "how to prove d(x,y) is metric space?" This is absurd, d(x,y) is a number, not a space. // Which axioms defining a metric space are a problem? $\endgroup$ – Did Sep 27 '13 at 14:23
  • $\begingroup$ soory i corrected it $\endgroup$ – Raja Sekar Sep 27 '13 at 14:27
  • $\begingroup$ Which axioms defining a metric space are a problem? $\endgroup$ – Did Sep 27 '13 at 14:29
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Hint: Suppose $X_1 = X_2 = \mathbb{R}$. You know how to make a metric on $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$ out of the metric on $\mathbb{R}$ in a way that looks like the one above (think: Pythagorean theorem). Try and generalize this.

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