Is this Hellinger Distance expression correct? Further to my previous post on the Hellinger Distance, there was one comment raised about there being different expressions of the Hellinger Distance. This has intrigued me.
In the Encyclopedia of Distances, 3rd Ed (p266), the Hellinger Distance is given as:

(I have used a picture to show the equation as given in the above reference)
However, I can’t get this to equate. Where have I gone wrong...assuming I have?
So from the above reference we have...the left-hand expression is:
$$\left[\sum_x \left (\sqrt{p_1 (x)}-\sqrt{p_2 (x)}\right)^{2}\right]^\frac{1}{2}$$
Re-writing as a radical
$$\sqrt{\sum_x \left (\sqrt{p_1 (x)}-\sqrt{p_2 (x)}\right)^{2}}$$
Expanding the square
$$\sqrt{\sum_x \left(\sqrt{p_1(x)}\right)^2-2\sqrt{p_1(x)}\sqrt{p_2(x)}+\left(\sqrt{p_2(x)}\right)^2}$$
Simplifying the radicals
$$\sqrt{\sum_x {p_1(x)}-2\sqrt{p_1(x)p_2(x)}+{p_2(x)}}$$
Sum of a probability distribution equals 1, so $\sum_x p(x)=1$;
$$\sqrt{\sum_x 1-2\sqrt{p_1(x)p_2(x)}+1}$$
Simplifying
$$\sqrt{\sum_x 2-2\sqrt{p_1(x)p_2(x)}}$$
Factorising
$$\sqrt{2\left(\sum_x 1-\sqrt{p_1(x)p_2(x)}\right)}$$
Moving the constant outside the radical and the minuend outside the summation
$$\sqrt2\sqrt{1-\sum_x \sqrt{p_1(x)p_2(x)}}$$
So I get a constant $\sqrt{2}$. That is okay as I am aware some expressions of the Hellinger distance have a bound of $[0,\sqrt{2}]$. However, I can't see where a constant of $2$ comes from as given in the above reference (which would give a bound of $[0,2]$).
Any help appreciated.

From $[\rm I], $

So, it's a typo.

Reference:
$\rm [I]$ Encyclopedia of Distances, Michel Marie Deza, Elena Deza, Springer-Verlag Berlin Heidelberg, $2016, $ p. $268.$
