We show that if a family of $\mathcal{A}$-harmonic functions that admits a
common growth condition is closed
in
$L^p_{\operatorname{loc}}$,
then this family is 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 $\mathcal{A}$-harmonic
functions
is finite dimensional if and only if it is closed in
$L^p_{\operatorname{loc}}$. 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.