  We introduce new vanishing subspaces of the homogeneous H\"{o}lder space
$\dot{C}^{0,\omega}(X,Y),$ in the generality of a doubling modulus $\omega$ and
normed spaces $X$ and $Y.$ For many couples $X,Y,$ we show these vanishing
subspaces to completely characterize those H\"older functions that admit
approximations, in the H\"{o}lder seminorm, by smooth, Lipschitz and boundedly
supported functions. Beyond the intrinsic interest of these results, we also
present connections to bi-parameter harmonic analysis on the Euclidean space
and in particular to compactness of the bi-commutator.
