Transfer of Approximation properties under Local Constraints and Best Simultaneous Approximation on Sums

Syamantak Das

arxiv: 2505.10851 · v3 · submitted 2025-05-16 · 🧮 math.FA

Transfer of Approximation properties under Local Constraints and Best Simultaneous Approximation on Sums

Syamantak Das This is my paper

Pith reviewed 2026-05-22 15:23 UTC · model grok-4.3

classification 🧮 math.FA

keywords approximation theoryBanach spacesM-idealssubspace sumssimultaneous approximation(GC) propertybest approximationfunctional analysis

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The pith

When the sum of two subspaces in a Banach space is closed, approximation properties transfer from one subspace to the sum if that subspace has the (GC) property.

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

This paper investigates the transfer of approximation properties between subspaces and their sums in Banach spaces. It focuses on cases where the sum is closed and one subspace possesses the (GC) property, showing that various related properties then hold for the sum. A sympathetic reader would care because these properties relate to best approximations and simultaneous approximations, which are fundamental in functional analysis. The authors provide counterexamples showing that some properties, including (GC) itself and being a central subspace, do not transfer. They resolve an open problem from a 2015 paper and extend the results to the properties (P1) and F-SACP.

Core claim

The central claim is that for closed sums of subspaces, the (GC) property in one implies certain approximation properties in the sum, while counterexamples disprove transfer for (GC) and central subspaces. The work answers the open problem raised in the 2015 paper on best constrained approximation and extends the study of best simultaneous approximations to include (P1) and F-SACP properties.

What carries the argument

The (GC) property, which combines two key approximation properties, acting as a condition under which properties transfer to closed sums of subspaces.

Load-bearing premise

The sum of the two subspaces is closed, which is required to discuss the transfer of properties such as (GC).

What would settle it

A concrete counterexample would be two subspaces A and B with A + B closed, A having the (GC) property, but the sum failing to have one of the expected approximation properties like best simultaneous approximation.

read the original abstract

It is folklore that the sum of two $M$-ideals (semi $M$-ideals) is also an $M$-ideal (a semi $M$-ideal). Numerous authors have attempted to investigate such properties of subspaces. This article explores two important facets of approximation theory within Banach spaces and how these properties remain intact when considering the sum of two subsets. Recall the notion of $(GC)$ introduced by Vesel\'y that encloses two aforementioned properties. When the sum of two subspaces is closed, we discuss various properties of the sum if one of the subspaces has these properties. Counterexamples are produced that establish nonaffirmativeness for the properties $(GC)$ and the central subspace. We answer a problem raised by the author in [{\em Best constrained approximation in Banach spaces}, Numer. Funct. Anal. Optim. {\bf 36}(2) (2015), 248--255]. We extend our observations related to the best simultaneous approximations to the properties $(P_1)$ and $\mr{F}$-SACP.

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

The paper supplies counterexamples showing non-transfer of (GC) and central subspace properties, plus a resolution of the 2015 open problem, but the counterexamples' relation to the closed-sum hypothesis needs explicit checking.

read the letter

The main takeaway is that this paper gives counterexamples establishing that the (GC) property and the central subspace property do not pass to the closed sum of two subspaces, and it resolves the open problem from the 2015 paper on best constrained approximation. It also extends the observations to (P1) and F-SACP. These are concrete additions to the existing literature on subspace properties in Banach spaces. The positive transfer results when the sum is closed follow the standard setup from M-ideals and Veselý's (GC) notion without introducing unnecessary new machinery. The work stays focused on what holds or fails under the closed-sum condition for the affirmative parts. The counterexamples are presented as showing nonaffirmativeness, which is a direct way to bound the claims. The stress-test concern about whether those counterexamples actually satisfy the closed-sum condition is worth a close look in the full text. If the constructions avoid closed sums, they demonstrate failure only outside the main hypothesis rather than inside it, which would weaken their impact on the transfer discussion. The abstract separates the closed-sum positive results from the counterexamples, so the paper needs to make that boundary clear to keep the resolution of the open problem sharp. This is a specialized piece for people already working on approximation properties and M-ideals in Banach spaces. A reader tracking subspace sums or simultaneous approximation will pick up the new examples and the settled question quickly. It is narrow but targets a stated open issue with explicit constructions. I would send it to peer review. The resolution and counterexamples give it enough substance for referees to evaluate the details and any needed clarifications around the hypotheses.

Referee Report

1 major / 2 minor

Summary. The paper explores transfer of approximation properties such as (GC), central subspace, (P1), and F-SACP to the closed sum of two subspaces in Banach spaces when one subspace possesses the property. It produces counterexamples establishing that (GC) and the central subspace property do not transfer in general, resolves an open problem posed in the 2015 reference on best constrained approximation, and extends observations on best simultaneous approximation to the properties (P1) and F-SACP.

Significance. If the derivations and counterexamples hold, the work clarifies the conditions under which approximation properties are preserved under closed sums of subspaces, directly resolving a 2015 open problem and broadening results to simultaneous approximation settings. The separation of positive transfer results (under closed sums) from general counterexamples is a useful contribution to the literature on M-ideals and related properties in functional analysis.

major comments (1)

[Abstract and counterexample sections] Abstract and counterexample sections: the counterexamples establishing non-transfer for (GC) and the central subspace property are presented separately from the closed-sum transfer discussion. It is not stated whether the sums in these counterexamples are closed. If the sums are not closed, the counterexamples demonstrate failure only outside the paper's primary closed-sum regime and do not address whether transfer holds (or fails) when the sum is closed; this distinction is load-bearing for the claimed resolution of the 2015 open problem and the extensions to (P1) and F-SACP, both framed within the closed-sum setting.

minor comments (2)

The abstract invokes 'folklore' regarding sums of M-ideals being M-ideals; adding a specific reference would strengthen the background.
Notation for F-SACP and (P1) is introduced without an explicit definition in the provided abstract; ensure these are defined at first use in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting this important point about the scope of our counterexamples. We address the comment below and will incorporate clarifications in a revised version.

read point-by-point responses

Referee: Abstract and counterexample sections: the counterexamples establishing non-transfer for (GC) and the central subspace property are presented separately from the closed-sum transfer discussion. It is not stated whether the sums in these counterexamples are closed. If the sums are not closed, the counterexamples demonstrate failure only outside the paper's primary closed-sum regime and do not address whether transfer holds (or fails) when the sum is closed; this distinction is load-bearing for the claimed resolution of the 2015 open problem and the extensions to (P1) and F-SACP, both framed within the closed-sum setting.

Authors: We agree that the closed or non-closed status of the sums in the counterexamples must be stated explicitly, as this affects how the results are interpreted relative to the closed-sum regime. In the counterexamples we construct for the failure of (GC) and the central subspace property to transfer, the sums of the subspaces are not closed. These examples are designed to demonstrate that the properties do not hold for sums in general, thereby underscoring the necessity of the closed-sum hypothesis in our positive transfer results for (P1) and F-SACP. The resolution of the 2015 open problem on best constrained approximation is presented strictly under the closed-sum assumption and does not rely on these counterexamples. To remove any potential ambiguity, we will revise the abstract and the counterexample sections to explicitly note that the sums are not closed in those constructions and to more clearly delineate the general non-transfer results from the closed-sum transfer theorems. This change will be implemented in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes transfer results for approximation properties such as (GC), (P1), and F-SACP when the sum of subspaces is closed and one subspace possesses the property, while separately constructing counterexamples showing non-transfer in general and resolving an open problem posed in the author's own 2015 paper. These steps consist of standard theorem proofs, explicit counterexample constructions, and direct answers to prior questions rather than any self-definitional equivalence, fitted parameter renamed as prediction, or load-bearing reduction to an unverified self-citation. The closed-sum hypothesis is an explicit assumption framing the positive results, and the counterexamples are presented as establishing non-affirmativeness outside that regime; no equation or claim reduces by construction to its own inputs or prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates within standard Banach space theory and approximation theory; no free parameters, ad-hoc axioms, or invented entities are apparent from the abstract.

axioms (2)

standard math Standard properties of M-ideals and semi M-ideals in Banach spaces as background folklore.
Invoked in the opening sentence of the abstract as established fact.
domain assumption The (GC) property introduced by Veselý combines two approximation properties.
Used as the central notion whose transfer is studied.

pith-pipeline@v0.9.0 · 5707 in / 1307 out tokens · 31912 ms · 2026-05-22T15:23:56.883714+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

When the sum of two subspaces is closed, we discuss various properties of the sum if one of the subspaces has these properties. Counterexamples are produced that establish nonaffirmativeness for the properties (GC) and the central subspace.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We answer a problem raised by the author in [Best constrained approximation in Banach spaces...]. We extend our observations related to the best simultaneous approximations to the properties (P1) and F-SACP.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Pradipta Bandyopadhyay, T. S. S. R. K. Rao ,Central subspaces of Banach spaces, J. Approx. Theory103(2) (2000), 206–222

work page 2000
[2]

Dutta,Weighted Chebyshev centers and intersection properties of balls in Banach spaces, Function spaces (Edwardsville, IL, 2002), 43–58, Contemp

Pradipta Bandyopadhyay, S. Dutta,Weighted Chebyshev centers and intersection properties of balls in Banach spaces, Function spaces (Edwardsville, IL, 2002), 43–58, Contemp. Math.,328, Amer. Math. Soc., Providence, RI, 2003

work page 2002
[3]

Baronti and P

M. Baronti and P. L. Papini,Norm-one projections onto subspaces of finite codimen- sion inl 1 andc 0, Period. Math. Hungar.22(3) (1991), 161–174

work page 1991
[4]

Optimization,75(1) (2024), 149–167

Syamantak Das, Tanmoy Paul,A study on various generalizations of generalized centers(GC)in Banach spaces. Optimization,75(1) (2024), 149–167

work page 2024
[5]

Syamantak Das and Tanmoy Paul,A study onF-simultaneous approximativeτ- compactness property in Banach spaces, Funct. Anal. Appl.60(1) (2026), 1–13

work page 2026
[6]

Diestel, J

J. Diestel, J. J. Uhl,Vector measures, Mathematical Surveys15, American Mathe- matical Society, Providence, R.I., (1977)

work page 1977
[7]

Harmand, D

P. Harmand, D. Werner, W. Werner,M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics1547, Springer-Verlag, Berlin, (1993)

work page 1993
[8]

James,Weak compactness and reflexivity, Israel J

Robert C. James,Weak compactness and reflexivity, Israel J. Math.2(1964), 101– 119

work page 1964
[9]

Lima,Intersection properties of balls and subspaces in Banach spaces, Trans

˚A. Lima,Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc.227(1977), 1–62. 18 DAS

work page 1977
[10]

Lima,,Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math

˚A. Lima,,Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math. Scand.53(1) (1983), 97–113

work page 1983
[11]

P. K. Lin,K¨ othe-Bochner function spaces, Birkh¨ auser Boston, Boston, MA, 2004

work page 2004
[12]

Lindenstruass ,Extension of compact operators

J. Lindenstruass ,Extension of compact operators. Mem. Amer. Math. Soc.48(1964)

work page 1964
[13]

Lindenstrauss,On projections with norm1−an example, Proc

J. Lindenstrauss,On projections with norm1−an example, Proc. Amer. Math. Soc. 15(1964), 403–406

work page 1964
[14]

Parthasarathy,Selection theorems and their applications, Lecture Notes in Math- ematics,263

T. Parthasarathy,Selection theorems and their applications, Lecture Notes in Math- ematics,263. Springer-Verlag, Berlin-New York, 1972

work page 1972
[15]

R. R. Phelps,Convex functions, monotone operators and differentiability, Second edition, Lecture Notes in Mathematics, 1364, Springer, Berlin, 1993

work page 1993
[16]

I. A. Pyatyshev,Operations over approximatively compact sets, Math. Notes82(5-6) (2007), 653–659

work page 2007
[17]

T. S. S. R. K. Rao,On intersections of ranges of projections of norm one in Banach spaces, Proc. Amer. Math. Soc.141(10) (2013), 3579–3586

work page 2013
[18]

T. S. S. R. K. Rao,Best constrained approximation in Banach spaces, Numer. Funct. Anal. Optim.36(2) (2015), 248–255

work page 2015
[19]

Walter Rudin,Functional Analysis, Second International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991

work page 1991
[20]

Libor Vesel´ y,Generalized centers of finite sets in Banach spacesActa Math. Univ. Comenian. (N.S.)66(1) (1997), 83–115

work page 1997
[21]

Libor Vesel´ y,Chebyshev centers in hyperplanes ofc 0, Czechoslovak Math. J. 52(127)(4) (2002), 721–729

work page 2002
[22]

Libor Vesel´ y,Polyhedral direct sums of Banach spaces, and generalized centers of finite sets, J. Math. Anal. Appl.391(2) (2012), 466–479. Syamantak Das,National Institute of Technology Sikkim, India E-mail address, Syamantak Das:syamantakdas@nitsikkim.ac.in

work page 2012

[1] [1]

Pradipta Bandyopadhyay, T. S. S. R. K. Rao ,Central subspaces of Banach spaces, J. Approx. Theory103(2) (2000), 206–222

work page 2000

[2] [2]

Dutta,Weighted Chebyshev centers and intersection properties of balls in Banach spaces, Function spaces (Edwardsville, IL, 2002), 43–58, Contemp

Pradipta Bandyopadhyay, S. Dutta,Weighted Chebyshev centers and intersection properties of balls in Banach spaces, Function spaces (Edwardsville, IL, 2002), 43–58, Contemp. Math.,328, Amer. Math. Soc., Providence, RI, 2003

work page 2002

[3] [3]

Baronti and P

M. Baronti and P. L. Papini,Norm-one projections onto subspaces of finite codimen- sion inl 1 andc 0, Period. Math. Hungar.22(3) (1991), 161–174

work page 1991

[4] [4]

Optimization,75(1) (2024), 149–167

Syamantak Das, Tanmoy Paul,A study on various generalizations of generalized centers(GC)in Banach spaces. Optimization,75(1) (2024), 149–167

work page 2024

[5] [5]

Syamantak Das and Tanmoy Paul,A study onF-simultaneous approximativeτ- compactness property in Banach spaces, Funct. Anal. Appl.60(1) (2026), 1–13

work page 2026

[6] [6]

Diestel, J

J. Diestel, J. J. Uhl,Vector measures, Mathematical Surveys15, American Mathe- matical Society, Providence, R.I., (1977)

work page 1977

[7] [7]

Harmand, D

P. Harmand, D. Werner, W. Werner,M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics1547, Springer-Verlag, Berlin, (1993)

work page 1993

[8] [8]

James,Weak compactness and reflexivity, Israel J

Robert C. James,Weak compactness and reflexivity, Israel J. Math.2(1964), 101– 119

work page 1964

[9] [9]

Lima,Intersection properties of balls and subspaces in Banach spaces, Trans

˚A. Lima,Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc.227(1977), 1–62. 18 DAS

work page 1977

[10] [10]

Lima,,Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math

˚A. Lima,,Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math. Scand.53(1) (1983), 97–113

work page 1983

[11] [11]

P. K. Lin,K¨ othe-Bochner function spaces, Birkh¨ auser Boston, Boston, MA, 2004

work page 2004

[12] [12]

Lindenstruass ,Extension of compact operators

J. Lindenstruass ,Extension of compact operators. Mem. Amer. Math. Soc.48(1964)

work page 1964

[13] [13]

Lindenstrauss,On projections with norm1−an example, Proc

J. Lindenstrauss,On projections with norm1−an example, Proc. Amer. Math. Soc. 15(1964), 403–406

work page 1964

[14] [14]

Parthasarathy,Selection theorems and their applications, Lecture Notes in Math- ematics,263

T. Parthasarathy,Selection theorems and their applications, Lecture Notes in Math- ematics,263. Springer-Verlag, Berlin-New York, 1972

work page 1972

[15] [15]

R. R. Phelps,Convex functions, monotone operators and differentiability, Second edition, Lecture Notes in Mathematics, 1364, Springer, Berlin, 1993

work page 1993

[16] [16]

I. A. Pyatyshev,Operations over approximatively compact sets, Math. Notes82(5-6) (2007), 653–659

work page 2007

[17] [17]

T. S. S. R. K. Rao,On intersections of ranges of projections of norm one in Banach spaces, Proc. Amer. Math. Soc.141(10) (2013), 3579–3586

work page 2013

[18] [18]

T. S. S. R. K. Rao,Best constrained approximation in Banach spaces, Numer. Funct. Anal. Optim.36(2) (2015), 248–255

work page 2015

[19] [19]

Walter Rudin,Functional Analysis, Second International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991

work page 1991

[20] [20]

Libor Vesel´ y,Generalized centers of finite sets in Banach spacesActa Math. Univ. Comenian. (N.S.)66(1) (1997), 83–115

work page 1997

[21] [21]

Libor Vesel´ y,Chebyshev centers in hyperplanes ofc 0, Czechoslovak Math. J. 52(127)(4) (2002), 721–729

work page 2002

[22] [22]

Libor Vesel´ y,Polyhedral direct sums of Banach spaces, and generalized centers of finite sets, J. Math. Anal. Appl.391(2) (2012), 466–479. Syamantak Das,National Institute of Technology Sikkim, India E-mail address, Syamantak Das:syamantakdas@nitsikkim.ac.in

work page 2012