  We introduce a new concept of perturbation of closed linear subspaces and
operators in Banach spaces called uniform lambda-adjustment which is weaker
than perturbations by small gap, operator norm, q-norm, and K2-approximation.
In arbitrary Banach spaces some of the classical Fredholm stability theorems
remain true under uniform lambda-adjustment, while other fail. However,
uniformly lambda-adjusted subspaces and linear operators retain their
(semi--)Fredholm properties in a Banach space which dual is Fr\'{e}chet-Urysohn
in weak* topology. We also introduce another concept of perturbation called
uniform mu-approximation which is weaker than perturbations by small gap, norm,
and compact convergence, yet stronger than uniform lambda-adjustment. We
present Fredholm stability theorems for uniform mu-approximation in arbitrary
Banach spaces and a theorem on stability of Riesz kernels and ranges for
commuting closed essentially Kato operators. Finally, we define the new
concepts of a tuple of subspaces and of a complex of subspaces in Banach
spaces, and present stability theorems for index and defect numbers of Fredholm
tuples and complexes under uniform lambda-adjustment and uniform
mu-approximation.
