Sard's theorem for mappings between Fr\'echet manifolds

TL;DR

This paper extends Sard's theorem to infinite-dimensional Fréchet manifolds, showing that for certain Lipschitz-Fredholm maps, the set of regular values is residual, thus generalizing classical finite-dimensional results.

Contribution

It proves an infinite-dimensional Sard's theorem for bounded Fréchet manifolds with Lipschitz-Fredholm maps, a significant generalization of the classical theorem.

Findings

Regular values form a residual set in the target manifold.

The theorem applies to bounded Fréchet manifolds with specific metric properties.

It requires the map to be an MC^k-Lipschitz-Fredholm map with k exceeding the index.

Abstract

In this paper we prove an infinite-dimensional version of Sard's theorem for Fr\'{e}chet manifolds. Let $ M $ and $ N $ be bounded Fr\'{e}chet manifolds such that the topologies of their model Fr\'{e}chet spaces are defined by metrics with absolutely convex balls. Let $ f: M \rightarrow N $ be an $ MC^k$-Lipschitz-Fredholm map with $ k > \max \lbrace {\Ind f,0} \rbrace $. Then the set of regular values of $ f $ is residual in $ N $.