Let X and Y be Banach
spaces, P : X → Y a Gateaux differentiable operator having closed graph.
Suppose
for each R > 0 there is a δ > 0 such that
P−1(K) is bounded whenever cl(K) ⊆ Y is compact;
then P is an open mapping of X onto Y . Similar results are obtained for compact
Gateaux differentiable operators using a local version of (i); the same local version
gives a domain invariance theorem for Gateaux differentiable operators having
closed graph. Related results deal with M. Altman’s theory of contractor
directions and theory of normal solvability as developed by F. E. Browder and
others.
