Editing Hilbert's fifth problem (section)

== Solution ==
The first major result was that of [[John von Neumann]] in 1933,<ref>{{cite journal|last=John|first=von Neumann|title=Die Einführung analytischer parameter in topologischen Gruppen|journal=Annals of Mathematics |year=1933 |volume=34 |pages=170–190 |doi=10.2307/1968347|issue=1|jstor=1968347}}</ref> giving an affirmative answer for [[compact group]]s. The [[locally compact abelian group]] case was solved in 1934 by [[Lev Pontryagin]]. The final resolution, at least in the interpretation of what Hilbert meant given above, came with the work of [[Andrew Gleason]], [[Deane Montgomery]] and [[Leo Zippin]] in the 1950s.

In 1953, [[Hidehiko Yamabe]] obtained further results about topological groups that may not be manifolds:{{efn|According to {{harvtxt|Morikuni|1961|p=i}}, "the final answer to Hilbert’s Fifth Problem"; however this is not so clear since there have been other such claims, based on different interpretations of Hilbert's statement of the problem given by various researchers. For a review of such claims (ignoring the contributions of Yamabe) see {{harvtxt|Rosinger|1998|pp=xiii–xiv and pp. 169–170}}}}

{{block indent|Every locally compact [[connected space|connected]] group is the [[projective limit]] of a sequence of Lie groups. Further, it is a Lie  group if it has no small subgroups.}}

It follows that every locally compact group contains an open subgroup that is a projective limit of Lie groups, by [[Totally disconnected group|van Dantzig's theorem]] (this last statement is called the Gleason–Yamabe Theorem in {{harvtxt|Tao|2014|loc=Theorem 1.1.17}}).