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When considering a metric for the infinite cartesian product $\mathbb{R}^w$, in Munkres's Topology (2nd edition, p.124) mentions that:

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Why is it that $d(x,y)$ "does not always makes sense, for the series in question need not to converge"? Similarly, why $\rho(x,y)$ does not make sense either?

Also, why convergence matters, when defining such a metric?

Thanks!

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    $\begingroup$ If the series diverged, what would $d(\mathbf x,\mathbf y)$ be? $\endgroup$ – Tim Raczkowski Mar 23 '16 at 2:39
  • $\begingroup$ What would be the distance from $(0,0,0,\dots )$ to $(1,1,1,\dots)$? $\endgroup$ – zhw. Mar 23 '16 at 3:23
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Imagine that you are calculating the distance between $x= (0,0,\dots,0,\dots)$ and $y =(1,1,\dots,1,\dots)$.

$$ d(x, y)^2 = \sum_{i=0}^\infty (1-0)^2 = \sum_1^\infty 1.$$

That sum is infinite, so its a series, and it doesn't converge, because the partial sums are $1,2,3,\dots$ etc. You need convergence, otherwise you don't have a real number that you can call the distance.

Considering $\rho$, what happens for the distance between $(1,2,3, \dots)$ and $(0,0,0,\ldots)$? Is it a real number (which it needs to be to be a distance)?

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If the series does not converge that means the sum has no value associated with it. For example let $x_k = 1$ and $y_k = 0$. Then using $d(\mathbf{x},\mathbf{y})$ you have,

$\displaystyle \sum_{n=1}^\infty 1$

and that is not defined. The other metric can lead to the same issue if the sequences aren't bounded. An example is $x_k = k$ and $y_k = 0$ does not have a value with that metric because the natural numbers lack an upper bound. Thus, if you define a metric make sure your definition actually gives a value.

One technical note is you can choose to allow your metric to output positive infinity which would be a different way to eliminate the problem in this situation. A metric where infinity is a possible value has pretty much all the properties of the usual metric so it isn't a bad thing to consider allowing (just be aware of what you want the range of your metric to be). And the only way those metrics give you a problem is if the result is positive infinity.

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