Local compactness for families of A-harmonic functions

We show that if a family of A-harmonic functions that admits a common growth condition is closed in L-loc(P), then this family is to locally compact on a dense open set under a family of topologies, all generated by norms. This implies that when this family of functions is a vector space, then such a vector space of A-harmonic functions is finite dimensional if and only if it is closed in L-loc(P). We then apply our theorem to the family of all p-harmonic functions on the plane with polynomial growth at most d to show that this family is essentially small.