Cartesian Product of Metric Spaces

[MKR]

Hello everyone.

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

$$\tau_p((x_1,y_1), (x_2,y_2)) = (\rho(x_1,x_2)^p + \sigma(y_1,y_2)^p)^\frac{1}{p}$$

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

$$(\rho(x_1,x_2)^p + \sigma(y_1,y_2)^p)^\frac{1}{p} \leq \rho(x_1,x_2) + \sigma(y_1,y_2)$$

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 $$\leq \rho(x_1,x_3) + \rho (x_3,x_2) + \sigma(y_1,y_3) + \sigma(y_3,y_2)$$

and I'm stuck. Thanks in advance for your help.

 

[Hurkyl]

How did you prove the Euclidean metric on R² satisfies the triangle equality? Wouldn't the same proof work here?

 

[MKR]

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!