Analytic Mappings Between LB-spaces and Applications in   Infinite-Dimensional Lie Theory
  We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups:   (a) groups of germs of analytic diffeomorphisms around a compact set in a Banach space and   (b) unions of ascending sequences of Banach Lie groups. 
Rafael Dahmen