  For a general class of saddle point problems sharp estimates for
Babu\v{s}ka's inf-sup stability constants are derived in terms of the constants
in Brezzi's theory. In the finite-dimensional Hermitian case more detailed
spectral properties of preconditioned saddle point matrices are presented,
which are helpful for the convergence analysis of common Krylov subspace
methods. The theoretical results are applied to two model problems from optimal
control with time-periodic state equations. Numerical experiments with the
preconditioned minimal residual method are reported.
