Collective compactness, equicompactness and the degree of nondensifiability

The degree of nondensifiability (DND) quantifies, in a specific sense, the distance between a bounded subset of a metric space and the Peano Continua it contains. In this paper, we use the DND to establish a quantitative version of the concepts of collective compactness and equicompactness for families of bounded linear operators between Banach spaces. Specifically, we prove inequalities that relate the DND to these notions and through some examples, we demonstrate that these inequalities are the best possible