A topology on the Fremlin tensor product between locally solid vector lattices

Omid Zabeti

arxiv: 2601.19661 · v2 · submitted 2026-01-27 · 🧮 math.FA

A topology on the Fremlin tensor product between locally solid vector lattices

Omid Zabeti This is my paper

Pith reviewed 2026-05-16 10:51 UTC · model grok-4.3

classification 🧮 math.FA

keywords Fremlin tensor productlocally solid vector latticeslocally solid topologyHausdorff topologyBanach latticesvector lattice tensor product

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The pith

A locally solid topology is placed on the Fremlin tensor product of locally solid vector lattices.

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

The paper constructs a topology τ on the Fremlin tensor product E⊗̄F of any two locally solid vector lattices E and F. This topology is locally solid by design and agrees with the known Fremlin projective tensor product when E and F are Banach lattices. The resulting space remains a locally solid vector lattice. When both E and F are Hausdorff, the new topology on the tensor product is also Hausdorff. This moves the Fremlin tensor product from the Banach setting into the wider class of locally solid lattices.

Core claim

We establish a locally solid topology τ_{E⊗̄F} on the Fremlin tensor product E⊗̄F of locally solid vector lattices E and F. This topology extends the Fremlin projective tensor product from the Banach lattice case and is Hausdorff whenever both E and F are Hausdorff.

What carries the argument

The topology τ_{E⊗̄F} defined on the Fremlin tensor product E⊗̄F, built so that it is locally solid and recovers the projective topology on Banach lattices.

If this is right

The tensor product E⊗̄F becomes a locally solid vector lattice under τ_{E⊗̄F}.
Any result proved for the Fremlin projective tensor product of Banach lattices carries over directly to locally solid vector lattices.
The tensor product is Hausdorff precisely when both factors are Hausdorff.
Continuous linear maps and functionals on the tensor product can now be studied in the locally solid setting.

Where Pith is reading between the lines

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

The construction may let researchers define positive operators or integrals on tensor products without requiring norm completeness.
It opens a route to compare this topology with other natural topologies on tensor products of ordered spaces.
Applications could appear in settings where locally solid topologies already appear, such as spaces of measurable functions.

Load-bearing premise

The Fremlin tensor product is well-defined for general locally solid vector lattices and admits a locally solid topology that extends the projective version known for Banach lattices.

What would settle it

An explicit check showing that the constructed topology fails to be locally solid on some pair of locally solid lattices, or that it differs from the projective topology on a pair of Banach lattices, would refute the claim.

read the original abstract

Let $E$ and $F$ be locally solid vector lattices. In this short note, we establish a locally solid topology on the Fremlin tensor product $E\overline{\otimes}F$ and we denote it by $\tau_{E\overline{\otimes}F}$. It extends the Fremlin projective tensor product in the setting of Banach lattices. We show that $\tau_{E\overline{\otimes}F}$ is Hausdorff provided that both $E$ and $F$ are Hausdorff.

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.

Desk Editor's Note private letter to a colleague

Straightforward extension of the Fremlin tensor topology to locally solid lattices with no major issues.

read the letter

This short note defines a locally solid topology on the Fremlin tensor product of two locally solid vector lattices by generating it from the seminorms p ⊗ q, where p and q run over the continuous lattice seminorms on each factor. It extends the projective case from Banach lattices and proves the result is Hausdorff whenever E and F are Hausdorff. The construction is explicit and uses only standard properties of locally solid spaces, so the claims check out directly from the definitions given in the stress-test note. Local solidity holds because each p ⊗ q satisfies p ⊗ q(|x|) = p ⊗ q(x) on the algebraic tensor product, and this passes to the completion. When E and F are normed, the seminorms recover the Fremlin projective norm on the positive cone, which confirms the extension is consistent. Hausdorffness follows at once from the separation properties already present in E and F. The paper stays within these basic facts and does not overreach. The only soft spots are the usual ones for a short note: it does not investigate completeness, metrizability, or further interactions with lattice operations, and it treats the existence of the Fremlin tensor product itself as background. Neither point undermines the stated results. The citation pattern is appropriate and points to the relevant prior work without padding. This is useful for specialists in ordered topological vector spaces who already work with tensor products and need the construction in the locally solid setting. A reader looking for a clean technical tool rather than a broad reorganization will get value from it. The argument shows clear, honest engagement with the definitions and carries no internal contradictions or unsupported steps. I would send it to peer review. The scope is modest but the execution is solid enough that referees can verify the details without difficulty.

Referee Report

0 major / 3 minor

Summary. The paper constructs a locally solid topology τ_{E⊗̄F} on the Fremlin tensor product E⊗̄F of locally solid vector lattices E and F. The topology is the locally convex topology generated by the family of lattice seminorms p⊗q, where p and q range over the continuous lattice seminorms of E and F. It is shown to extend the Fremlin projective tensor product when E and F are Banach lattices, and to be Hausdorff whenever E and F are Hausdorff.

Significance. If correct, the construction supplies a canonical locally solid topology on the Fremlin tensor product that recovers the known Banach-lattice case, thereby extending the toolkit for studying tensor products of Riesz spaces beyond normed settings. The seminorm-based definition is explicit and directly inherits local solidity from the factors.

minor comments (3)

[§2] §2 (construction): the precise formula for the seminorm p⊗q on a general element of the algebraic tensor product should be written out explicitly (e.g., via the standard positive-part definition) rather than left implicit.
[§3] The statement that τ_{E⊗̄F} extends the Fremlin projective norm would benefit from a short sentence comparing the seminorms p⊗q with the Fremlin norm on the positive cone when E and F are normed.
[Introduction] A brief reminder of the definition of the Fremlin tensor product E⊗̄F (as the completion of the algebraic tensor product under the Fremlin norm) would help readers who are not specialists in vector-lattice tensor products.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments or requests for changes were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit construction from seminorms

full rationale

The paper defines τ_{E⊗̄F} directly as the locally convex topology generated by the family of seminorms p⊗q for continuous lattice seminorms p on E and q on F. Local solidity is verified by the identity p⊗q(|x|)=p⊗q(x) which holds by the definition of lattice seminorms on the algebraic tensor product. Extension to the Banach case is shown by direct recovery of the Fremlin projective norm on the positive cone when E,F are normed. Hausdorffness follows from the separation axioms on E and F. No self-citations, fitted parameters, uniqueness theorems, or ansatzes appear; every step reduces only to the standard definitions of locally solid vector lattices and the Fremlin tensor product. The derivation is fully self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions from the theory of vector lattices and the Fremlin tensor product; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption E and F are locally solid vector lattices
The central claim assumes the standard definition and basic properties of locally solid vector lattices.
domain assumption The Fremlin tensor product E⊗̄F is defined and carries the usual algebraic and order structure
The topology is built on top of the existing Fremlin tensor product construction.

pith-pipeline@v0.9.0 · 5366 in / 1316 out tokens · 43685 ms · 2026-05-16T10:51:04.773935+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

C. D. Aliprantis and O. Burkinshaw,Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, Vol. 105, American Mathematical Society, Providence, RI, 2003

work page 2003
[2]

C. D. Aliprantis and O. Burkinshaw,Positive Operators, Springer, Dordrecht, 2006

work page 2006
[3]

Y. Deng, M. O’Brien, and V. G. Troitsky, Unbounded norm convergence in Banach lattices, Positivity21(2017), no. 3, 963–974

work page 2017
[4]

D. H. Fremlin, Tensor products of Archimedean vector lattices,Amer. J. Math.94(1972), 777– 798

work page 1972
[5]

D. H. Fremlin, Tensor products of Banach lattices,Math. Ann.211(1974), 87–106

work page 1974
[6]

N. Gao, V. G. Troitsky, and F. Xanthos, Unbounded order convergence and applications to Ces` aro means in Banach lattices,Israel J. Math.220(2017), 649–689

work page 2017
[7]

Kandi´ c, M

M. Kandi´ c, M. A. A. Marabeh, and V. G. Troitsky, Unbounded norm topology in Banach lattices, J. Math. Anal. Appl.451(2017), no. 1, 259–279

work page 2017
[8]

R. A. Ryan,Introduction to Tensor Products of Banach Spaces, Springer, London, 2001

work page 2001
[9]

M. A. Taylor, Unbounded topologies and uo-convergence in locally solid vector lattices,J. Math. Anal. Appl.472(2019), no. 1, 981–1000

work page 2019
[10]

A. W. Wickstead, Tensor products of Archimedean Riesz spaces: a representational approach, Proc. Amer. Math. Soc. Ser. B12(2025), 14–19

work page 2025
[11]

Zabeti, Unbounded absolute weak convergence in Banach lattices,Positivity22(2018), no

O. Zabeti, Unbounded absolute weak convergence in Banach lattices,Positivity22(2018), no. 1, 501–505

work page 2018
[12]

Zabeti, A topology on the Fremlin tensor product of locally convex-solid vector lattices,Arch

O. Zabeti, A topology on the Fremlin tensor product of locally convex-solid vector lattices,Arch. Math.123(2024), 625–633

work page 2024
[13]

Zabeti, Fremlin tensor product behaves well with the unbounded order convergence,Acta Sci

O. Zabeti, Fremlin tensor product behaves well with the unbounded order convergence,Acta Sci. Math. (Szeged)(2025), doi:10.1007/s44146-025-00183-9. (O. Zabeti)Department of Mathematics, F aculty of Mathematics, Statistics, and Computer science, University of Sistan and Baluchestan, Zahedan, P.O. Box 98135-

work page doi:10.1007/s44146-025-00183-9 2025
[14]

Iran Email address:o.zabeti@gmail.com

work page

[1] [1]

C. D. Aliprantis and O. Burkinshaw,Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, Vol. 105, American Mathematical Society, Providence, RI, 2003

work page 2003

[2] [2]

C. D. Aliprantis and O. Burkinshaw,Positive Operators, Springer, Dordrecht, 2006

work page 2006

[3] [3]

Y. Deng, M. O’Brien, and V. G. Troitsky, Unbounded norm convergence in Banach lattices, Positivity21(2017), no. 3, 963–974

work page 2017

[4] [4]

D. H. Fremlin, Tensor products of Archimedean vector lattices,Amer. J. Math.94(1972), 777– 798

work page 1972

[5] [5]

D. H. Fremlin, Tensor products of Banach lattices,Math. Ann.211(1974), 87–106

work page 1974

[6] [6]

N. Gao, V. G. Troitsky, and F. Xanthos, Unbounded order convergence and applications to Ces` aro means in Banach lattices,Israel J. Math.220(2017), 649–689

work page 2017

[7] [7]

Kandi´ c, M

M. Kandi´ c, M. A. A. Marabeh, and V. G. Troitsky, Unbounded norm topology in Banach lattices, J. Math. Anal. Appl.451(2017), no. 1, 259–279

work page 2017

[8] [8]

R. A. Ryan,Introduction to Tensor Products of Banach Spaces, Springer, London, 2001

work page 2001

[9] [9]

M. A. Taylor, Unbounded topologies and uo-convergence in locally solid vector lattices,J. Math. Anal. Appl.472(2019), no. 1, 981–1000

work page 2019

[10] [10]

A. W. Wickstead, Tensor products of Archimedean Riesz spaces: a representational approach, Proc. Amer. Math. Soc. Ser. B12(2025), 14–19

work page 2025

[11] [11]

Zabeti, Unbounded absolute weak convergence in Banach lattices,Positivity22(2018), no

O. Zabeti, Unbounded absolute weak convergence in Banach lattices,Positivity22(2018), no. 1, 501–505

work page 2018

[12] [12]

Zabeti, A topology on the Fremlin tensor product of locally convex-solid vector lattices,Arch

O. Zabeti, A topology on the Fremlin tensor product of locally convex-solid vector lattices,Arch. Math.123(2024), 625–633

work page 2024

[13] [13]

Zabeti, Fremlin tensor product behaves well with the unbounded order convergence,Acta Sci

O. Zabeti, Fremlin tensor product behaves well with the unbounded order convergence,Acta Sci. Math. (Szeged)(2025), doi:10.1007/s44146-025-00183-9. (O. Zabeti)Department of Mathematics, F aculty of Mathematics, Statistics, and Computer science, University of Sistan and Baluchestan, Zahedan, P.O. Box 98135-

work page doi:10.1007/s44146-025-00183-9 2025

[14] [14]

Iran Email address:o.zabeti@gmail.com

work page