If a structure group G is a complex semisimple matrix group, the Sobolev completion G ¯ ¯ k ( X ) of a gauge group G ( X ) can be introduced. It is a Lie group. A key point is that the action of G ¯ ¯ k ( X ) on a Sobolev completion A k of a space of principal connections is smooth, and that an orbit space A k / G ¯ ¯ k ( X ) is a Hilbert space. It is a configuration space of quantum gauge theory.