  In this paper, we prove that the notions of Hilbert stability and Mumford
stability agree for vector bundles of arbitrary rank over smooth curves. The
notion of Hilbert stability was introduced by Gieseker and Morrison in 1984,
and they showed that for smooth curves and vector bundles of rank two it agrees
with Mumford stability. A different proof for the rank two case was given by M.
Teixidor i Bigas. Our proof uses a new approach and avoids complicated
computations. Our results might serve as a first step in the construction of
the Hilbert stable compactification of the universal moduli space of stable
vector bundles over the moduli space of smooth curves as suggested by Teixidor.
