P-Product Metric is Metric

Theorem
Let $$M_{1'} = \left\{{A_{1'}, d_{1'}}\right\}, M_{2'} = \left\{{A_{2'}, d_{2'}}\right\}, \ldots, M_{n'} = \left\{{A_{n'}, d_{n'}}\right\}$$ be metric spaces.

Let $$\mathcal {M} = \left\{{\prod_{i=1}^n \left\{{A_{i'}, d_{i'}}\right\}, d_n}\right\}$$, where the definition of $$d_n$$ is defined as:

$$ $$ $$

where $$x = \left({x_1, x_2, \ldots, x_n}\right) \in \prod_{i=1}^n A_{i'}$$ and $$y = \left({y_1, y_2, \ldots, y_n}\right) \in \prod_{i=1}^n A_{i'}$$.

These product spaces are metrics.

Proof for p = 1
This is the taxicab metric, which has been proved to be a metric.

Proof for p = 2, 3, ...
It is easy to see that conditions M0, M1 and M2 of the conditions for being a metric are satisfied. So all we need to do is check M3.

Let:
 * 1) $$z = \left({z_1, z_2, \ldots, z_n}\right)$$;
 * 2) all summations be over $$i = 1, 2, \ldots, n$$;
 * 3) $$d_{i'} \left({x_i, y_i}\right) = r_i$$;
 * 4) $$d_{i'} \left({y_i, z_i}\right) = s_i$$.

Then:

$$\left({\sum \left({d_{i'} \left({x_i, y_i}\right)}\right)^p}\right)^{\frac 1 p} + \left({\sum \left({d_{i'} \left({y_i, z_i}\right)}\right)^p}\right)^{\frac 1 p} = \left({\sum r_i^p}\right)^{\frac 1 p} + \left({\sum s_i^p}\right)^{\frac 1 p}$$.

As all the $$d_{i'}$$ are metrics, $$d_{i'} \left({x_i, y_i}\right) \ge 0$$ and $$d_{i'} \left({y_i, z_i}\right) \ge 0$$.

Thus $$r_i = \left|{r_i}\right|$$ and $$s_i = \left|{s_i}\right|$$, and of course $$r_1 + s_i = \left|{r_i + s_i}\right|$$.

This reduces our condition for being a metric space to:

$$\left({\sum \left|{r_i}\right|^p}\right)^{\frac 1 p} + \left({\sum \left|{s_i}\right|^p}\right)^{\frac 1 p} \ge \left({\sum \left|{r_i + s_i}\right|^p}\right)^{\frac 1 p}$$

This is Minkowski's Inequality.

Proof for Infinite Case
We have that $$d_\infty \left({x, y}\right) = \max_{i=1}^n \left\{{d_{i'} \left({x_i, y_i}\right)}\right\}$$.

Let $$k \in \left[{1 \,. \, . \, n}\right]$$ such that $$d_{k'} \left({x_k, y_k}\right) = d_\infty \left({x, z}\right) = \max_{i=1}^n \left\{{d_{i'} \left({x_i, y_i}\right)}\right\}$$.

Then by the definition of a metric, $$d_{k'} \left({x_k, z_k}\right) \le d_{k'} \left({x_k, y_k}\right) + d_{k'} \left({y_k, z_k}\right)$$.

But by the nature of the $$\max$$ operation, $$d_{k'} \left({x_k, y_k}\right) \le \max_{i=1}^n \left\{{d_{i'} \left({x_i, y_i}\right)}\right\}$$ and $$d_{k'} \left({y_k, z_k}\right) \le \max_{i=1}^n \left\{{d_{i'} \left({y_i, z_i}\right)}\right\}$$

Thus $$d_{i'} \left({x_i, y_i}\right) + d_{k'} \left({y_k, z_k}\right) \le \max_{i=1}^n \left\{{d_{i'} \left({x_i, y_i}\right)}\right\} + \max_{i=1}^n \left\{{d_{i'} \left({y_i, z_i}\right)}\right\}$$.

Hence $$d_\infty \left({x, z}\right) \le d_\infty \left({x, y}\right) + d_\infty \left({y, z}\right)$$.

Comment on notation
It can be shown that $$d_\infty \left({x, y}\right) = \lim_{r \to \infty} d_r \left({x, y}\right)$$.

That is, $$\lim_{r \to \infty} \left({\sum_{i=1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^r}\right)^{\frac 1 r} = \max_{i=1}^n \left\{{d_{i'} \left({x_i, y_i}\right)}\right\}$$.

Hence the notation.