On the inverse limit of finite dimensional lie algebras

We extend the well Known Levi-Malcev decomposition theorem of finite dimensional Lie algebras to the case of pro-finite dimensional Lie algebras L = limLn (n ∈ N). We also prove that every finite dimensional homomorphic image of the Cartesian product of finite dimensional nilpotent Lie algebras is also nilpotent.

On the closed subspaces of the inverse limit of finite dimensional vector spaces

We prove that the sum A + B of closed subspaces A and B of the inverse limit of finite dimensional vector spaces, V = limVn (n ∈ N) over an algebraically closed field of characteristic 0 is closed. We extend also the basic fact that every ideal of a finite dimensional semisimple Lie algebra has a unique complement to the case of closed ideals of prosemisimple Lie algebras.