# Monotonic bijection from positive reals to reals between 0 and 1

While engaging in personal study with regards to metric spaces I wished to construct a metric space of sequences. As a part of this I need a function that normalized positive reals into the range between 0 and 1. The idea being that I can define my metric as.

$d(x,y) = \sum_{n=1}^\infty f( |x_n - y_n| )10^{-n}$

I don't see any reason why such a function shouldn't exist but I cannot think of one.

• You could use something resembling the function $1/x$. Commented Jan 29, 2016 at 4:39
• What do you mean by "isomorphism"?
– user228113
Commented Jan 29, 2016 at 4:39
• @G.Sassatelli Likely an order isomorphism. Commented Jan 29, 2016 at 4:39

$f(x) = 1 - \frac{1}{1 + x}$.

There are multiple functions that can do this with slight modifications, including arctan and 1/x.

Shifting $R^+$ to $R^{>1}$ by $x\to x+1$ will work for $1/x$, giving us $\frac{1}{x+1}$

Arctan just needs to be scaled to fit in the range: $\frac{2}{\pi}\arctan(x)$

• I thought of 1/X but 1/X is outside my codomian when X< 1 Commented Jan 29, 2016 at 4:41
• That's what the composition I mentioned is for. $x\to x+1$ moves R^+ to R>1, so then you can apply it. I'll edit to make this clearer Commented Jan 29, 2016 at 4:43
• @QthePlatypus is that clearer Commented Jan 29, 2016 at 4:48

$$\quad\frac2\pi\arctan\quad$$