  The paper develops the theory of topological radicals of Banach Lie algebras
and studies the structure of Banach Lie algebras with sufficiently many Lie
subalgebras of finite codimensions -- the intersection of all these subalgebras
is zero. It is shown that the intersections of certain families of Lie
subalgebras (closed Lie subalgebras of finite codimension, closed Lie ideals of
finite codimension, closed maximal Lie subalgebras of finite codimension,
closed maximal Lie ideals of finite codimension) correspond to different
preradicals, and that these preradicals generate the same radical, the Frattini
radical. The main attention is given to structural properties of
Frattini-semisimple Banach Lie algebras and, in particular, to a new
infinite-dimensional phenomenon associated with the strong Frattini preradical
introduced in this paper. A constructive description of Frattini-free Banach
Lie algebras is obtained.
