  In this paper we prove a new abstract stability result for perturbed
saddle-point problems based on a norm fitting technique. We derive the
stability condition according to Babuska's theory from a small inf-sup
condition, similar to the famous Ladyzhenskaya-Babuska-Brezzi (LBB) condition,
and the other standard assumptions in Brezzi's theory, in a combined abstract
norm. The construction suggests to form the latter from individual fitted norms
that are composed from proper seminorms. This abstract framework not only
allows for simpler (shorter) proofs of many stability results but also guides
the design of parameter-robust norm-equivalent preconditioners. These benefits
are demonstrated on mixed variational formulations of generalized Poisson,
Stokes, vector Laplace and Biot's equations.
