Before revision:  New results include a characterization of convergence with respect to Wasserstein metrics, and the existence of barycenters (Fr \'e chet means ) for any distribution of diagrams  .
After revision:  New results include a characterization of convergence with respect to Wasserstein metrics, and the existence of barycenters (Fr \'e chet means ) for any distribution of diagrams , and an exhaustive description of continuous linear representations of persistence diagrams .