  We prove that given an analytic action of a compact $p$-adic Lie group on a
Banach space over a field of positive characteristic, one can detect either the
simultaneous vanishing or the simultaneous finite-dimensionality of all of the
continuous cohomology groups from the corresponding statement for the
restriction to a pro-$p$ procyclic subgroup. We also formulate a conjecture
generalizing this result, in which the base field is allowed to have mixed
characteristic and the subgroup is allowed to be nilpotent. Finally, we
formulate an analogous conjecture about Lie algebra cohomology and relate this
to a theorem of Kostant.
