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arxiv: 2503.07023 · v2 · submitted 2025-03-10 · 🧮 math.CA · math.DG· math.FA

On spaces of arc-smooth maps

Armin Rainer This is my paper

Pith reviewed 2026-05-23 01:08 UTC · model grok-4.3

classification 🧮 math.CA math.DGmath.FA
keywords arc-smooth mapsbornological isomorphismsfunction spacesexponential lawsconvenient vector spacesHölder boundaryultradifferentiable classes
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The pith

Arc-smooth functions on suitable closed sets coincide with smooth functions via bornological isomorphism of their natural topologies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for closed sets in Euclidean space satisfying Hölder boundary conditions or the fat subanalytic topological condition, the collection of arc-smooth maps equals the usual smooth maps not only setwise but as bornological spaces. This equivalence respects the locally convex topologies that arise naturally on these function spaces. The result extends directly to maps taking values in convenient vector spaces, producing exponential laws that identify mapping spaces with ordinary function spaces. Parallel statements hold for the Braun-Meise-Taylor ultradifferentiable classes.

Core claim

The author proves that the set-theoretic identities between arc-smooth and smooth function spaces, previously obtained on closed sets with Hölder boundary or fat subanalytic structure, are in fact bornological isomorphisms with respect to the natural locally convex topologies. Extending the setting to maps valued in convenient vector spaces yields corresponding exponential laws. Analogous bornological isomorphisms are shown for the special ultradifferentiable Braun-Meise-Taylor classes.

What carries the argument

Bornological isomorphism between the arc-smooth and smooth function spaces equipped with their natural locally convex topologies.

If this is right

  • Exponential laws identify spaces of maps into convenient vector spaces with ordinary function spaces on these closed domains.
  • The same bornological isomorphisms hold for the Braun-Meise-Taylor ultradifferentiable classes.
  • Calculus constructions that rely on the bornological structure transfer unchanged from open domains to these closed sets.
  • Set-theoretic descriptions of function spaces on such domains can be used directly in the category of convenient vector spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result supplies a route to define differentiable structures on domains that are not open while retaining the full strength of exponential laws.
  • It suggests checking whether other classes of maps, such as those with controlled loss of derivatives, admit similar bornological identifications on the same closed sets.
  • One could test whether the isomorphism persists when the target space is replaced by a more general locally convex space without the convenient property.

Load-bearing premise

The closed sets must already satisfy the Hölder boundary or fat subanalytic topological conditions that make arc-smoothness equivalent to ordinary smoothness.

What would settle it

An explicit closed set obeying the Hölder or fat subanalytic condition together with a sequence of functions whose bornological seminorms differ in the arc-smooth versus standard smooth topologies.

read the original 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 H\"older 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that for closed sets X in R^d satisfying Hölder-boundary or fat-subanalytic conditions (with the necessary topological condition), the set-theoretic equalities C^∞(X,Y)=C^∞_arc(X,Y) and analogous identities for real-analytic and ultradifferentiable classes induce bornological isomorphisms under the natural locally convex topologies. It extends the results to maps into convenient vector spaces to obtain exponential laws and proves parallel statements for Braun-Meise-Taylor classes.

Significance. If the results hold, the work supplies a topological (bornological) strengthening of earlier set-theoretic characterizations of smoothness on non-open domains. The derivation of exponential laws is a clear strength, as it directly supports categorical constructions in convenient calculus. The extension to Braun-Meise-Taylor classes broadens the scope without introducing new ad-hoc parameters.

minor comments (3)
  1. Introduction: the precise statements of the main theorems should restate the topological conditions on X (Hölder boundary or fat subanalytic) that are required for the underlying set equalities, even though they are inherited from prior work.
  2. Preliminaries: the bornology on the function spaces C^∞(X,E) should be defined explicitly before the isomorphism statements, to make the verification that the identification preserves the bornology fully transparent.
  3. References: ensure the bibliography contains complete citations to the prior papers establishing the set-theoretic arc-smooth characterizations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recognition of the bornological strengthening, exponential laws, and extension to Braun-Meise-Taylor classes. The recommendation of minor revision is noted.

Circularity Check

1 steps flagged

Minor self-citation for prerequisite set equalities; bornological isomorphism proof remains independent

specific steps
  1. self citation load bearing [Abstract]
    "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."

    The central claim applies the bornological-isomorphism property to identities whose validity is granted solely by the author's own prior work; without independent verification of those identities the new statement has no domain of application, though the isomorphism proof itself is not shown to reduce to the citation.

full rationale

The paper explicitly relies on its own prior work for the set-theoretic identities C^∞(X,Y) = C^∞_arc(X,Y) etc. under Hölder-boundary or fat-subanalytic conditions, then proves these identities induce bornological isomorphisms and exponential laws. This self-citation supplies a prerequisite assumption rather than reducing the new topological claims to a fit, definition, or unverified loop; the isomorphism argument is presented as a separate derivation building on the granted equalities. No equations in the provided text show the bornological result collapsing into the prior characterization by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results in convenient calculus and the topological conditions on the sets established in the author's previous work; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The sets under consideration satisfy the Hölder-boundary or fat-subanalytic topological conditions required for the arc-smooth characterization to hold.
    Invoked in the abstract as the setting in which the new isomorphism results are proved.

pith-pipeline@v0.9.0 · 5645 in / 1284 out tokens · 27705 ms · 2026-05-23T01:08:34.656164+00:00 · methodology

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Reference graph

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