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On spaces of arc-smooth maps

Armin Rainer

TL;DR

This work extends arc-smooth (Boman-type) characterizations of smooth and analytic maps from open domains to closed, tame sets such as simple fat closed subanalytic and Hölder sets. It proves that arc-smooth function spaces coincide with their classical counterparts (e.g., $ ext{AC}^ullet = ext{C}^ullet$) and that the resulting identities are bornological isomorphisms, with exponential laws enabling currying for vector-valued maps. The paper also develops the real-analytic and ultradifferentiable Braun–Meise–Taylor frameworks, establishing when arc-analytic or arc-ultradifferentiable spaces coincide with standard spaces under robust weight conditions, and it extends these results to vector-valued settings and two-dimensional plot extensions. Collectively, these results unify open and closed-domain analysis in the convenient setting, provide robust exponential laws, and offer tools for analysis on singular/tame spaces with applications to o-minimal structures and beyond.

Abstract

It is well-known that a function on an open set in $\mathbb R^d$ is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real analytic version) to suitable closed sets, notably, sets with Hölder boundary and fat subanalytic sets satisfying a necessary topological condition. In this paper, we prove that the resulting set-theoretic identities of function spaces are bornological isomorphisms with respect to their natural locally convex topologies. Extending the results to maps with values in convenient vector spaces, we obtain corresponding exponential laws. Additionally, we show analogous results for special ultradifferentiable Braun-Meise-Taylor classes.

On spaces of arc-smooth maps

TL;DR

This work extends arc-smooth (Boman-type) characterizations of smooth and analytic maps from open domains to closed, tame sets such as simple fat closed subanalytic and Hölder sets. It proves that arc-smooth function spaces coincide with their classical counterparts (e.g., ) and that the resulting identities are bornological isomorphisms, with exponential laws enabling currying for vector-valued maps. The paper also develops the real-analytic and ultradifferentiable Braun–Meise–Taylor frameworks, establishing when arc-analytic or arc-ultradifferentiable spaces coincide with standard spaces under robust weight conditions, and it extends these results to vector-valued settings and two-dimensional plot extensions. Collectively, these results unify open and closed-domain analysis in the convenient setting, provide robust exponential laws, and offer tools for analysis on singular/tame spaces with applications to o-minimal structures and beyond.

Abstract

It is well-known that a function on an open set in is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real analytic version) to suitable closed sets, notably, sets with Hölder boundary and fat subanalytic sets satisfying a necessary topological condition. In this paper, we prove that the resulting set-theoretic identities of function spaces are bornological isomorphisms with respect to their natural locally convex topologies. Extending the results to maps with values in convenient vector spaces, we obtain corresponding exponential laws. Additionally, we show analogous results for special ultradifferentiable Braun-Meise-Taylor classes.

Paper Structure

This paper contains 28 sections, 42 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convenient vector spaces and $c^\infty$-topology
  4. Uniform boundedness principle
  5. Hölder sets
  6. Subanalytic sets
  7. Smooth maps
  8. Arc-smooth functions
  9. The space $\mathcal{C}^\infty(X)$
  10. Arc-smooth functions have smooth extensions
  11. The vector valued case
  12. Exponential laws
  13. Real analytic maps
  14. The space $\mathcal{AC}^\omega(X)$
  15. The space $\mathcal{C}^\omega(X)$
  16. ...and 13 more sections

Key Result

Lemma 2.1

[l]l:stab Let $\mathcal{F}$ be a set of bounded linear maps $f:E\to E_f$ between locally convex spaces, let $\mathcal{S}_f$ be a point separating set of bounded linear maps on $E_f$ for every $f\in \mathcal{F}$, and let $\mathcal{S}:= \{g\circ f: f\in \mathcal{F}, g\in \mathcal{S}_f\}$. If $\mathcal

Theorems & Definitions (90)

  • Lemma 2.1: KM97
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 80 more