Topological radicals, IV. Frattini theory of Banach Lie algebras

We develop the theory of topological radicals in Banach Lie algebras, consider various ways of constructing such radicals and study a family of (pre)radicals related to subalgebras and ideals of finite codimension in Banach Lie algebras. It extends to the infinite-dimensional case many results of the well known Frattini theory of Lie algebras.