Metric induced by norm (cartesian product)

Question

$(X,d_X), (Y,d_Y)$ are metric Spaces and $Z=X×Y$ the cartesian product of $X$ and $Y$.

It's $d_Z:Z×Z \to \mathbb{R_{\geq0}}$ with $d_Z=\sqrt{d_X^2+d_Y^2}$

$d_X$ and $d_Y$ are induced by norms $||·||_X$ on $X$ and $||·||_Y$ on $Y$.

How to prove that $d_Z$ is also induced by a norm on $Z$?

I tried with $d(x,y) = ||x-y||$ but i didn't find a solution.

Answer 1

Just compute $d_Z((x,y), (x’,y’))$ using the given formula and make the obvious guess.

Hint:

Verify that $$\|(u,v)\|_{Z} = \sqrt{\|u\|_X ^2 + \|v\|_Y ^2}$$

is a norm on $Z$ that induces $d_Z$.