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I am given a set of sequences $S = \{(x_n)_n | x_n \in \{0, 1\}\}$ and the following function: $d(x, y) = \sum_n \frac{|x_n - y_n|}{2^n}, x, y \in S.$

Noting that the series is strictly positive, as is the fraction, are the following statements correct?

  1. $d(x,x) = 0 \iff |x_n - x_n| = 0$
  2. $d(x, y) = d(y, x) \iff |x_n - y_n| = |y_n - x_n|$
  3. $d(x, y) + d(y, z) \ge d(x, z) \iff |x_n - y_n| + |y_n - z_n| \ge |x_n - z_n|$
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    See also: http://math.stackexchange.com/questions/472516/how-to-prove-that-this-series-is-a-metric-dx-y-sum-i-0-infty-fracx-i2016-04-01

1 Answers 1

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I would do minor changes:

  1. $d(x,x) = 0 \iff |x_n - x_n| = 0$; If you used the implication $(\forall n\in\mathbb N)|x_n - x_n| = 0$ $\Rightarrow$ $d(x,x)=0$ instead, it would seem a little clearer to me.

  2. $d(x, y) = d(y, x) \iff |x_n - y_n| = |y_n - x_n|$; Again I would write this as $(\forall n\in\mathbb N) |x_n - y_n| = |y_n - x_n|$ $\Rightarrow$ $d(x, y) = d(y, x)$

  3. $d(x, y) + d(y, z) \ge d(x, z) \iff |x_n - y_n| + |y_n - z_n| \ge |x_n - z_n|$; the same thing here.

In the first two cases it is perhaps not so important, but in the third one the implication $\Longrightarrow$ is not true.


In case you have already learned something about products of topological spaces, you will find out that this is in fact the Cantor space. This metric is explicitly mentioned e.g. in Banach Space Theory: The Basis for Linear and Nonlinear Analysis By Marián Fabian, Petr Habala et al., p.739


BTW is there a name for this metric? It is neither Fréchet metric $d_F$ nor Baire metric $d_B$. $d_F(x,y)=\sum_{n=1}^\infty\frac1{2^n}\frac{|x_n-y_n|}{1+|x_n-y_n|}$ $d_B(x,y)=\frac1{\min\{n\in\mathbb N; x_n\ne y_n\}} \qquad\text{for }x\ne y$ These two metrics are used frequently for sequences space and their natural generalizations can be used to define topology on countable product of metric spaces.