sentences:  A Gårding domain for "U" is a linear subspace of "H" that is "U"("g")- and "U"("δ")-invariant for all "g" ∈ "G" and "δ" ∈ g and is also a domain of essential self-adjointness for "U"
  
  Gårding showed in 1947 that, if "G" is a Lie group, then a Gårding domain for "U" consisting of infinitely differentiable vectors exists for each continuous unitary representation of "G".In 1961, Kats extended this result to arbitrary locally compact topological groups.However, these results do not extend easily to the non-locally compact case because of the lack of a Haar measure on the group.