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Let $X$ be a set equipped with a metric $d_x$, denoted by $\langle X,d_x\rangle$, and $Y$ equipped with a metric $d_y$, denoted by $\langle X,d_y\rangle$. Let $Z=X\times Y$.

Let $z_1=(x_1,y_1), \ z_2=(x_2,y_2), \ \forall z_1,z_2\in Z$, we define $d_p$ and $d_{\infty}$ by:

$(i) \ d_p(z_1,z_2)=(d_x(x_1,x_2)^p+d_y(y_1,y_2))^p)^{1/p}$, for $p \in \{1,2\}$ $(ii) \ d_\infty(z_1,z_2)=\max\{d_x(x_1,x_2),d_y(y_1,y_2)\}.$

Are $\langle Z,d_p\rangle$ and $\langle Z,d_\infty\rangle$ metric spaces?"

I have already shown that $d_p$ for $p=1$ is a distance function, the other two cases are now left (for $p=2$ and $d_{\infty}$).

I have already shown that all properties of a distance function are fullfilled except for the triangle inequality. How do I show this?

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By Minkowski inequality, $d_p$ satisfies triangle inequality if $1\le p \le \infty$. It is easy that $d_p$ satisfies another conditions of definition of metric.

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