Compactness properties of the integration map for Volterra measures in non-normable spaces

We make a detailed study of the compactness properties of the integration map associated with the classical Volterra (vector valued) measure inX=L ^p(0, 1]), forp in 1, ∞], and inX=C(0, 1]), whereX is equipped with its relevant weak or weak-star topology. The spaceX (a Frechet space) of all complex valued sequences, equipped with the pointwise convergence topology, is also considered.