What does `n` correspond to here? And why is it “ours”? (although the second I understand as euclidean space corresponds to n=2 and we seem to live in a locally euclidean space)
Surely it’s not dimensions, since all of these examples were two-dimensional (x and y). So I’m a little lost here.
Sorry, on rereading this I've actually written a bunch of nonsense. It's true that each choice of n gives you back a metric, but the |x|, |y|, |z| actually refer to the components of your vectors. So in 1-dim space you just have |x|, in 2-dim space you have |x| and |y|, 3-dim space has |x|, |y| and |z| and so forth.
So in 3-dimensional space, for n=3, the "length" of a vector u=(u_x, u_y, u_z) is: d(u) = (|u_x|^3 + |u_y|^3 + |u_z|^3)^(1/3)
In 2-dimensional space, for n=4 and u=(u_x, u_y) you get the following: d(u) = (|u_x|^4 + |u_y|^4)^(1/4)
If you want to use the norm as a metric to work out the "distance" between two vectors u and v then you just compute d(u-v) where u-v is ordinary vector subtraction.
It's an infinite family of metrics - you provide an n (a positive integer) and get back a metric.
So if you pick n=1 you get d(x, y) = |x| + |y|, which is the taxicab metric. You can apply this metric to a Euclidean space of whatever dimension you like, just substituting the appropriate definition of |x| and |y|. For 1-dimensional space you would use |x| = abs(x[0]), for 2-dimensional space you would use |x| = sqrt(x[0]**2 + x[1]**2), etcetera. Hope that helps.