  Necessary and sufficient conditions for the exponentiation of
finite-dimensional real Lie algebras of linear operators on complete Hausdorff
locally convex spaces are obtained, focused on the equicontinuous case - in
particular, necessary conditions for exponentiation to compact Lie groups are
established. Applications to complete locally convex algebras, with special
attention to locally C$^*$-algebras, are given. The definition of a projective
analytic vector is introduced, playing an important role in some of the
exponentiation theorems.
