Classic Formulation
A formulation that was accepted for a long period was that the question was to characterize Lie groups as the topological groups that were also topological manifolds. In terms closer to those that Hilbert would have used, near the identity element e of the group G in question, we have some open set U in Euclidean space containing e, and on some open subset V of U we have a continuous mapping
that satisfies the group axioms where those are defined. This much is a fragment of a typical locally Euclidean topological group. The problem is then to show that F is a smooth function near e (since topological groups are homogeneous spaces, they look the same everywhere as they do near e).
Another way to put this is that the possible differentiability class of F doesn't matter: the group axioms collapse the whole Ck gamut.
Read more about this topic: Hilbert's Fifth Problem
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