  We study a version of the BGG category O for Dynkin Borel subalgebras of
root-reductive Lie algebras g, such as gl(\infty). We prove results about
extension fullness and compute the higher extensions of simple modules by Verma
modules. In addition, we show that our category O is Ringel self-dual and
initiate the study of Koszul duality. An important tool in obtaining these
results is an equivalence we establish between appropriate Serre subquotients
of category O for g and category O for finite dimensional reductive subalgebras
of g.
