An integrability criterion for a projective limit of Banach distributions

TL;DR

This paper establishes an integrability criterion for projective limits of Banach distributions on Fréchet manifolds, extending Frobenius and Lie theorems to infinite-dimensional settings.

Contribution

It introduces a new integrability criterion for projective limits of Banach distributions, enabling the extension of classical theorems to Fréchet manifolds.

Findings

Provides an integrability criterion for projective limits of Banach distributions.

Extends Frobenius theorem to the Fréchet setting.

Derives a version of the third Lie theorem for Fréchet-Lie groups.

Abstract

We give an integrability criterion for a projective limit of Banach distributions on a Fr\'echet manifold which is a projective limit of Banach manifolds. This leads to a result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds. This can be seen as a version of Frobenius Theorem in Fr\'echet setting. As consequence, we obtain a version of the third Lie theorem for a Fr\'echet-Lie group which is a submersive projective limit of Banach Lie groups. We also give an application to a sequence of prolongations of a Banach Lie algebroid.