On the domain of implicit functions in a projective limit setting without additional norm estimates

TL;DR

This paper explores the existence of implicit functions on ILB-Fréchet spaces without relying on traditional metric or norm estimates, expanding the theoretical framework for inverse function theorems in infinite-dimensional analysis.

Contribution

It introduces a method to obtain implicit functions on non-open domains in ILB-Fréchet spaces without classical norm estimates, and discusses a generalized differentiation concept.

Findings

Implicit functions can be constructed without metric or norm estimates.

An inverse function theorem is established under these new conditions.

Open questions remain on the appropriate notion of differentiation for Frobenius theorems.

Abstract

We examine how implicit functions on ILB-Fr\'echet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain $D$ which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse functions theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fr\"obenius theorem.