Denote by Cs(X;E) the space of the continuous functions defined on the completely regular and Hausdorff space X, with values in the locally convex topological vector space E, when it is endowed with the simple or point-wise convergence topology. We give here some conditions on X and on E under which the space Cs(X;E) is bornological or ultrabornological and characterize in some cases the corresponding associated spaces. We give also a few results concerning the case of the compact connvergence topology.