Regularity in infinite dimensions
The theme of this chapter has origins in the early days of functional analysis and the Banach open mapping theorem, which concerns continuous linear mappings from one Banach space to another. The graphs of such mappings are subspaces of the product of the two Banach spaces, but remarkably much of the classical theory extends to set-valued mappings whose graphs are convex sets or cones instead of subspaces. Openness connects up then with metric regularity and interiority conditions on domains and ranges, as seen in the Robinson–Ursescu theorem. Infinite-dimensional inverse function theorems and implicit function theorems due to Lyusternik, Graves, and Bartle and Graves can be derived and extended. Banach spaces can even be replaced to some degree by more general metric spaces.