It is shown that a
topological group G is topologically isomorphic to the isometry group of a (complete)
metric space if and only if G coincides with its 𝒢δ-closure in the Raĭkov completion
of G (resp. if G is Raĭkov-complete). It is also shown that for every Polish
(resp. compact Polish; locally compact Polish) group G there is a complete
(resp. proper) metric d on X inducing the topology of X such that G is isomorphic
to Iso(X,d), where X = ℓ2 (resp. X = [0,1]ω; X = [0,1]ω∖{point}). It is
demonstrated that there are a separable Banach space E and a nonzero vector e ∈ E
such that G is isomorphic to the group of all (linear) isometries of E which leave the
point e fixed. Similar results are proved for arbitrary Raĭkov-complete topological
groups.
Keywords
Polish group, isometry group, Hilbert cube, Hilbert space,
Hilbert cube manifold, Raĭkov-complete group, isometry
group of a Banach space
