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Diffeological Spaces with a Non-Smooth Derivation

Masaki Taho

Abstract

We show that on certain diffeological spaces there exist linear derivations that satisfy the Leibniz rule but are not smooth with respect to the given diffeology. This reveals that the notion of tangent space defined via all such derivations is strictly larger than the one defined using only smooth derivations, showing that smoothness cannot be recovered from the Leibniz rule alone.

Diffeological Spaces with a Non-Smooth Derivation

Abstract

We show that on certain diffeological spaces there exist linear derivations that satisfy the Leibniz rule but are not smooth with respect to the given diffeology. This reveals that the notion of tangent space defined via all such derivations is strictly larger than the one defined using only smooth derivations, showing that smoothness cannot be recovered from the Leibniz rule alone.
Paper Structure (5 sections, 4 theorems, 33 equations)

This paper contains 5 sections, 4 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Background on diffeological spaces and tangent spaces
  3. Diffeological spaces
  4. The external tangent spaces and the right tangent spaces
  5. A diffeological space with non-smooth derivations

Key Result

Theorem 1.1

Let $X=\bigvee_{n\in\mathbb{N}} \mathbb{R}$ be the bouquet of countably many copies of $\mathbb{R}$ equipped with the quotient diffeology, and let $0\in X$ be the wedge point. Then we have isomorphisms of vector spaces In particular, Hence the external and right tangent spaces do not coincide.

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1: IZ
  • Definition 2.2: IZ
  • Example 2.3
  • Definition 2.4: IZ
  • Definition 2.5: IZ
  • Definition 2.6: IZ
  • Definition 2.7: IZ
  • Definition 2.8: IZ
  • Example 2.9: Irrational tori, IZ
  • ...and 11 more