An implicit function theorem for Banach spaces and some applications

TL;DR

This paper establishes a generalized implicit function theorem for Banach spaces without requiring complemented subspaces, and applies it to problems in Banach-Lie groups, including fiber parametrization, subgroup characterization, and local rigidity.

Contribution

It introduces a new implicit function theorem for Banach spaces that relaxes the complemented subspace assumption, enabling broader applications.

Findings

Generalized implicit function theorem proven for Banach spaces

Applied to parametrization of differentiable map fibers

Extended to Lie subgroup and local rigidity problems

Abstract

We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach-Lie groups, as well as Weil's local rigidity for homomorphisms from finitely generated groups to Banach-Lie groups.