  We describe basic diffeological structures related to splittings and
Grassmannians for infinite dimensional vector spaces. We analyze and expand the
notion of non-commutative cross-ratio and prove its smoothness. Then we
illustrate this theory by examples, with some of them extracted from the
existing literature related to infinite dimensional (Banach) Grassmannians, and
others where the diffeological setting is a key primary step for rigorous
definitions.
