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We study concepts of stability 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 results imply that stability of the dual span holds under a hypothesis related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability.Finally, using our results we obtain stable vector bundles of slope 3, and prove that they admit theta-divisors

LINEAR SERIES ON CURVES: STABILITY AND CLIFFORD INDEX