Diffeological Spaces with a Non-Smooth Derivation

TL;DR

This paper demonstrates that in certain diffeological spaces, there exist derivations satisfying Leibniz rule that are not smooth, indicating the tangent space defined via all derivations is larger than the smooth tangent space.

Contribution

It reveals that the tangent space in diffeological spaces cannot be fully characterized by smooth derivations alone, highlighting a fundamental distinction.

Findings

Existence of non-smooth derivations satisfying Leibniz rule

The tangent space from all derivations is strictly larger

Smoothness cannot be inferred solely from Leibniz rule

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.