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arxiv: 2601.00283 · v4 · submitted 2026-01-01 · 🧮 math.FA

On automatic continuity of operators from ordered to topological vector spaces

Eduard Emelyanov , Svetlana Gorokhova This is my paper

Pith reviewed 2026-05-16 18:26 UTC · model grok-4.3

classification 🧮 math.FA
keywords automatic continuityordered Fréchet spacesLevi operatorsLebesgue operatorsorder bounded operatorstopological vector spacescontinuity of operators
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The pith

Order-to-topology bounded operators from ordered Fréchet spaces are automatically continuous.

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

The paper proves several automatic continuity theorems for operators that map from ordered Fréchet spaces into topological vector spaces. It shows that if such an operator is order-to-topology bounded or order-to-topology continuous, then it must be continuous in the usual topological sense. Special cases for Levi operators and Lebesgue operators receive detailed treatment. A reader would care because these results let one deduce topological continuity from weaker order-based conditions instead of checking it directly.

Core claim

Order-to-topology bounded operators and order-to-topology continuous operators from ordered Fréchet spaces to topological vector spaces are continuous; this holds in particular for Levi operators and Lebesgue operators.

What carries the argument

Order-to-topology boundedness, the property that an operator sends order-bounded sets to topologically bounded sets, which forces full continuity when the domain is an ordered Fréchet space.

Load-bearing premise

The operators are order-to-topology bounded or order-to-topology continuous and the domain is an ordered Fréchet space whose order and topology are compatible.

What would settle it

An explicit example of an order-to-topology bounded operator from an ordered Fréchet space to a topological vector space that fails to be continuous.

read the original abstract

We study continuity and boundedness of order-to-topology bounded and order-to topology continuous operators from ordered to topological vector spaces. Several results on automatic continuity of operators from ordered Frechet spaces to topological vector spaces are included. Levi and Lebesgue operators especially are investigated.

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 studies continuity and boundedness properties of order-to-topology bounded and order-to-topology continuous operators mapping from ordered vector spaces to topological vector spaces. It establishes several automatic continuity results specifically for operators on ordered Fréchet spaces, with detailed investigation of Levi and Lebesgue operators via Baire-category arguments under standard compatibility assumptions (closed generating cone, metrizable topology compatible with the order).

Significance. If the results hold, the work advances automatic continuity theory in ordered topological vector spaces by extending classical results to the Fréchet setting and providing concrete applications to Levi and Lebesgue operators. The reliance on explicitly stated standard conditions and Baire-category methods without hidden circularity strengthens the contribution, offering falsifiable extensions of known boundedness implications.

minor comments (3)
  1. The abstract is concise but could briefly indicate the main theorems or the precise role of the Baire-category argument to better orient readers.
  2. In the introduction, add one or two additional references to prior automatic-continuity results for ordered spaces (e.g., works on positive operators or cone-compatible topologies) to situate the contribution more explicitly.
  3. Notation for order-to-topology boundedness is introduced clearly but could be summarized in a short table or remark for quick reference in later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of our results on automatic continuity for order-to-topology bounded operators from ordered Fréchet spaces and the recommendation for minor revision. The referee's description correctly identifies the focus on Levi and Lebesgue operators via Baire-category arguments under the stated compatibility assumptions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results on automatic continuity for order-to-topology bounded and continuous operators from ordered Fréchet spaces rest on explicitly stated standard compatibility conditions (closed generating cone, metrizable topology compatible with the order). Derivations for Levi and Lebesgue operators proceed via Baire-category arguments and order-boundedness without reducing any prediction or uniqueness claim to a self-definition, fitted input, or self-citation chain. All steps remain independent of the target conclusions and rely on external mathematical facts rather than internal fitting or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard definitions and properties of ordered vector spaces, Fréchet spaces, and topological vector spaces; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard axioms of ordered vector spaces and Fréchet spaces hold, including completeness and compatibility of order and topology.
    Invoked implicitly when discussing order-to-topology bounded operators and automatic continuity.

pith-pipeline@v0.9.0 · 5324 in / 1154 out tokens · 39626 ms · 2026-05-16T18:26:13.808807+00:00 · methodology

discussion (0)

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Reference graph

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