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Typical Lipschitz mappings are typically non-differentiable

Michael Dymond, Olga Maleva

TL;DR

The work establishes a category-based, extreme form of non-differentiability for typical Lipschitz mappings between Banach spaces. By leveraging a Banach–Mazur game framework and a local perturbation (gluing) technique, the authors show that for any separable subspace $W$ of $\mathcal{L}(X,Y)$, a dense $G_{\delta}$ set of 1-Lipschitz maps has a residual set of points $x$ (in a prescribed $G\subseteq \operatorname{Int} Q$) where every operator in $\mathbb{B}_{W}$ behaves like a derivative, i.e., $\mathcal{D}_{f}(x)\supseteq \mathbb{B}_{W}$. This implies that typical Lipschitz mappings are non-differentiable on a large, dense subset of the domain, even in finite-dimensional Euclidean cases, and strengthens previous results by showing maximal differentiation obstruction on a residual set. The results also address and refute related remarks in prior literature, highlighting the distinct behavior from measure-theoretic differentiability outcomes like Rademacher’s theorem. Overall, the paper advances our understanding of how differentiation can fail in a category-theoretic sense for Lipschitz maps beyond almost-everywhere statements.

Abstract

We prove that a typical Lipschitz mapping between any two Banach spaces is non-differentiable at typical points of any given subset of its domain in the most extreme form. This is a new result even for Lipschitz mappings between Euclidean spaces.

Typical Lipschitz mappings are typically non-differentiable

TL;DR

The work establishes a category-based, extreme form of non-differentiability for typical Lipschitz mappings between Banach spaces. By leveraging a Banach–Mazur game framework and a local perturbation (gluing) technique, the authors show that for any separable subspace of , a dense set of 1-Lipschitz maps has a residual set of points (in a prescribed ) where every operator in behaves like a derivative, i.e., . This implies that typical Lipschitz mappings are non-differentiable on a large, dense subset of the domain, even in finite-dimensional Euclidean cases, and strengthens previous results by showing maximal differentiation obstruction on a residual set. The results also address and refute related remarks in prior literature, highlighting the distinct behavior from measure-theoretic differentiability outcomes like Rademacher’s theorem. Overall, the paper advances our understanding of how differentiation can fail in a category-theoretic sense for Lipschitz maps beyond almost-everywhere statements.

Abstract

We prove that a typical Lipschitz mapping between any two Banach spaces is non-differentiable at typical points of any given subset of its domain in the most extreme form. This is a new result even for Lipschitz mappings between Euclidean spaces.
Paper Structure (8 sections, 8 theorems, 55 equations)

This paper contains 8 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Main result.
  3. Preliminaries and Notation.
  4. General notation and differentiability notions.
  5. Optimality of Theorem \ref{['thm:typical_nonsep']}.
  6. Lipschitz mappings.
  7. The Banach-Mazur Game
  8. Proof of Theorem \ref{['thm:typical_nonsep']}

Key Result

Theorem 1.1

Let $X$ and $Y$ be Banach spaces, $W$ be a separable subspace of $\mathcal{L}(X,Y)$, $Q$ be a closed and bounded subset of $X$ and let $G\subseteq \operatorname{Int} Q$. Then there is a residual subset $\mathcal{F}$ of $\operatorname{Lip}_{1}(Q,Y)$ such that for every $f\in \mathcal{F}$ the set is residual in $G$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • ...and 7 more