Unnormalized Hellinger distance ($h(P,Q)=\Vert \sqrt{p}-\sqrt{q}{\Vert }_{L^2(\nu )}$)

The Hellinger distance convention $h^2(P,Q)=\int (\sqrt{p}-\sqrt{q}{)}^{2}d\nu$, with no factor $1/2$, for densities under a common dominating measure. Its range is $[0,\sqrt{2}]$. This convention makes $\mathrm{TV}(P,Q)\le h(P,Q)\le \sqrt{\mathrm{KL}(P,Q)}$ hold with the displayed constants.