Definition talk:General Euclidean Metric

The literature that I have to hand specify the General Euclidean Metric for integer $r$ only. Intuitively it seems sound to extend to $\R$ and from what I can tell Minkowski's Inequality for Sums holds for real $r \ge 1$ ... am I okay to assume that my source work 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces is careless? (I would not be surprised - it's not the cleanest exposition I've seen.) --prime mover (talk) 22:03, 29 October 2012 (UTC)

Yes. Cf. Definition:P-Seminorm for a general treatment (in fact, coming to think of it, the general Euclidean metric is an instance of the metrics these norms induce - how general is measure theory). --Lord_Farin (talk) 22:40, 29 October 2012 (UTC)

"For applications in analysis," says Sutherland, p.32 op.cit., "metric spaces have a strong rival in normed vector spaces. The latter are particularly convenient for generalizing differential calculus and for handling linear problems in general ..."--prime mover (talk) 22:57, 29 October 2012 (UTC)

What makes this metric Euclidean?

As far as I understand matters, it's only preserved by Euclidean transformations for $r=2$, so why the name? This is only one of infinitely many families of metrics generating the usual topology for $\mathbb R^n$. --Dfeuer (talk) 03:23, 17 December 2012 (UTC)

It's not the euclidean metric. It's the generalized euclidean metric. --prime mover (talk) 06:22, 17 December 2012 (UTC)