Extremum Problems on a Hilbert Network

As a generalization of a usual infinite network, a Hilbert network is defined as a pair of a graph and a resistance taking values in a Hilbert space. With the sets of nodes and arcs of the graph, we associate variables belonging to a Hilbert space. In this situation, we study several extremum problems related to Hilbert-valued functions on the set of nodes or arcs of the graph and their inverse relations.