  We study concepts of stabilities associated to a smooth complex curve
together with a linear series on it. In particular we investigate the relation
between stability of the associated Dual Span Bundle and linear stability. Our
result implies a stability condition related to the Clifford index of the
curve. Furthermore, in some of the cases, we prove that a stronger stability
holds: cohomological stability. Eventually using our results we obtain stable
vector bundles of integral slope 3, and prove that they admit theta-divisors.
