Local compactness and closedness for families of <math> <?Pub Eqn> <f> <sc>A</sc></f> </math>-harmonic functions.

Abstract

We show that closed families of A -harmonic functions whose members all admit a common growth condition admits many topologies, all of which are generated by norms, so that for each of those topologies, tau, there is a set, <italic>U</italic>_tau, that is open, dense and locally compact under tau. When the family is a vector space this implies that the family is finite dimensional.