Metric of a product

**tttcomrader** · September 11th 2009, 03:43 PM

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!

**flyingsquirrel** · September 12th 2009, 12:24 AM

Hello,

Originally Posted by tttcomrader

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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