Definition:General Euclidean Metric

Let $$\reals^n$$ be an $n$-dimensional real vector space.

Let $$x = \left({x_1, x_2, \ldots, x_n}\right) \in \reals^n$$ and $$y = \left({y_1, y_2, \ldots, y_n}\right) \in \reals^n$$.

Consider the Euclidean metric $$d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{\frac 1 2}$$ on $$\reals^n$$.

The generalized Euclidean metrics are defined as follows:

$$ $$ $$

The Generalized Euclidean Metric is a Metric.

The Generalized Euclidean Metrics are Topologically Equivalent.

Note that $$d_2 \left({x, y}\right)$$ is the usual Euclidean metric.

Relationship with Product Space Metrics
It can be seen that this is a special case of a product space.

Note
To complete the family, we could also define $$d_0$$ as the standard discrete metric on $$\reals^n$$.

However, while $$d_1, d_2, \ldots, d_\infty$$ are all topologically equivalent, this is not the case with $$d_0$$.