  Let $U$ be a Banach Lie group and $G\le U$ a compact subgroup. We show that
closed Lie subgroups of $U$ contained in sufficiently small neighborhoods
$V\supseteq G$ are compact, and conjugate to subgroups of $G$ by elements close
to $1\in U$; this generalizes a well-known result of Montgomery and Zippin's
from finite- to infinite-dimensional Lie groups. Along the way, we also prove
an approximate counterpart to Jordan's theorem on finite subgroups of general
linear groups: finite subgroups of $U$ contained in sufficiently small
neighborhoods $V\supseteq G$ have normal abelian subgroups of index bounded in
terms of $G\le U$ alone.
  Additionally, various spaces of compact subgroups of $U$, equipped with the
Hausdorff metric attached to a complete metric on $U$, are shown to be analytic
Banach manifolds; this is the case for both (a) compact groups of a given,
fixed dimension, or (b) compact (possibly disconnected) semisimple subgroups.
Finally, we also prove that the operation of taking the centralizer (or
normalizer) of a compact subgroup of $U$ is continuous (respectively upper
semicontinuous) in the appropriate sense.
