  A holonomic space $(V,H,L)$ is a normed vector space, $V$, a subgroup, $H$,
of $Aut(V, \|\cdot\|)$ and a group-norm, $L$, with a convexity property. We
prove that with the metric $d_L(u,v)=\inf_{a\in
H}\{\sqrt{L^2(a)+\|u-av\|^2}\}$, $V$ is a metric space which is locally
isometric to a Euclidean ball. Given a Sasaki-type metric on a vector bundle
$E$ over a Riemannian manifold, we prove that the triplet $(E_p,Hol_p,L_p)$ is
a holonomic space, where $Hol_p$ is the holonomy group and $L_p$ is the length
norm defined within. The topology on $Hol_p$ given by the $L_p$ is finer than
the subspace topology while still preserving many desirable properties. Using
these notions, we introduce the notion of holonomy radius for a Riemannian
manifold and prove it is positive. These results are applicable to the
Gromov-Hausdorff convergence of Riemannian manifolds.
