Let G be a topological
group and let G∗ denote the space of all Lebesgue measurable functions from the unit
interval [0,1] into G with the topology of convergence in measure. With this
topology and with pointwise multiplication as the group operation, G∗ is
a topological group. If G is separable and has a complete metric and has
more than one point, then Bessaga and Pefczyński have shown that G∗ is
homeomorphic to l2, separable infinite-dimensional Hilbert space. This fact is used
in this paper to show the existence of separable Fréchet manifolds which
are topological groups and which have certain algebraic and topological
properties.
