arxiv: 2604.24529 · v1 · submitted 2026-04-27 · 🧮 math.FA

Recognition: unknown

About smooth and non-poor subspaces of Daugavet spaces

Samir Hamad

Pith reviewed 2026-05-07 17:27 UTC · model grok-4.3

classification 🧮 math.FA

keywords Daugavet propertyGâteaux differentiabilityMüntz spacesC[0,1]quotient spacesslice diameter two propertynon-complete normed spaces

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The pith

A non-complete normed space can have the Daugavet property with its norm Gâteaux differentiable at every nonzero point.

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

The paper constructs an example of a non-complete normed space that has the Daugavet property while its norm is Gâteaux differentiable at every nonzero vector. It notes that the dual norm on any normed space with the Daugavet property fails to be Gâteaux differentiable at any point. It further identifies quasilacunary Müntz spaces inside C[0,1] that are isomorphic to c0 and shows that the quotients by these subspaces lack the Daugavet property even though the slice diameter two property survives the construction. These distinctions clarify how completeness interacts with smoothness and different diameter-two properties in normed spaces.

Core claim

We discuss an example of a non-complete normed space with the Daugavet property such that the norm is Gâteaux differentiable at every nonzero point. In contrast, we note that the dual norm of a normed space with the Daugavet property is not Gâteaux differentiable at any point. Furthermore, we show that quasilacunary Müntz spaces form a natural class of subspaces of C[0,1], isomorphic to c0, for which the corresponding quotient spaces fail to have the Daugavet property. At the same time, the slice diameter two property is preserved under this construction.

What carries the argument

The explicit construction of a non-complete normed space that is simultaneously Daugavet and Gâteaux smooth at every nonzero point, together with the class of quasilacunary Müntz subspaces of C[0,1] that separate the Daugavet property from the slice diameter two property in the quotient.

If this is right

Completeness is required for the usual incompatibility between the Daugavet property and Gâteaux smoothness of the norm.
The slice diameter two property can hold in quotients even when the full Daugavet property does not.
Quasilacunary Müntz subspaces supply a concrete mechanism for producing quotients that distinguish these two properties.

Where Pith is reading between the lines

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

The non-complete example may serve as a test case for whether other smoothness notions or higher-order differentiability can coexist with the Daugavet property outside Banach spaces.
Similar constructions could be attempted in other classical function spaces to produce further separations between diameter-two properties.
The dual non-differentiability result might extend to show that Daugavet spaces are always non-smooth in their bidual or other natural enlargements.

Load-bearing premise

The particular construction produces a genuinely non-complete space that satisfies the Daugavet property while its norm remains Gâteaux differentiable at every nonzero point.

What would settle it

A direct verification that the constructed space is either complete or fails to be Gâteaux differentiable at some nonzero point, or that its dual norm is Gâteaux differentiable somewhere.

read the original abstract

We discuss an example of a non-complete normed space with the Daugavet property such that the norm is G\^ateaux differentiable at every nonzero point. In contrast, we note that the dual norm of a normed space with the Daugavet property is not G\^ateaux differentiable at any point. Furthermore, we show that quasilacunary M\"untz spaces form a natural class of subspaces of $C[0,1]$, isomorphic to $c_0$, for which the corresponding quotient spaces fail to have the Daugavet property. At the same time, the slice diameter two property is preserved under this construction.

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 gives an explicit non-complete smooth Daugavet space plus targeted quotient results on quasilacunary Müntz subspaces that lose Daugavet but keep slice diameter two.

read the letter

The main thing to know is that the authors build a concrete dense subspace of a Banach space, equip it with a custom norm, and verify directly that it has the Daugavet property while the norm is Gâteaux differentiable at every nonzero point. They also show that the dual norm on any Daugavet space fails to be Gâteaux differentiable anywhere, and they give a class of quasilacunary Müntz subspaces of C[0,1] (isomorphic to c0) whose quotients fail Daugavet but preserve the slice diameter two property. The verifications use the rank-one operator characterization for Daugavet and straightforward limit computations for differentiability, plus the standard Müntz criterion for the isomorphism and a direct slice estimate for the diameter-two preservation. This is hands-on, checkable work that fills in specific gaps without relying on heavy machinery. The soft spots are limited and expected for the area: the results stay tightly focused on these examples and do not claim wider consequences, so the paper will mainly interest people already working on Daugavet spaces and related subspace properties. No load-bearing gaps in the arguments show up from the details provided, and the constructions avoid circularity. This is for specialists in Banach space theory who track smoothness, Daugavet-type properties, and Müntz-type subspaces. A reader in that niche will find the explicit contrasts useful and the proofs straightforward to follow. It deserves a serious referee because the examples are new, the verifications are direct, and the claims are narrow enough to be settled by checking the constructions rather than broad interpretation.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs an explicit non-complete normed space equipped with a custom norm that satisfies the Daugavet property (verified via the rank-one operator formulation) while being Gâteaux differentiable at every nonzero vector (verified by direct computation of the directional derivative). It proves that the dual norm on any normed space with the Daugavet property fails to be Gâteaux differentiable at any point, via a short argument producing norm-attaining functionals. It further identifies quasilacunary Müntz subspaces of C[0,1] that are isomorphic to c0 (via the standard Müntz theorem criterion) and shows that the corresponding quotients fail the Daugavet property (by exhibiting a rank-one operator with norm strictly less than 1 + ||T||) while the slice diameter two property is preserved (via a direct slice-diameter estimate).

Significance. If the constructions hold, the work separates the Daugavet property from non-smoothness in the incomplete setting and supplies a concrete counterexample to any expectation that Daugavet spaces must be non-smooth. The dual-norm observation is a clean general fact. The quasilacunary Müntz examples provide a natural, explicitly describable class of c0-isomorphic subspaces of C[0,1] in which Daugavet fails for quotients but slice diameter two persists; the explicit verifications (rank-one operators, direct differentiability limits, and slice estimates) are strengths that make the claims falsifiable and checkable.

minor comments (3)

The title refers to 'non-poor subspaces' but the abstract and early sections do not define or gloss the term; a one-sentence explanation or forward reference would improve accessibility.
In the construction of the non-complete space (presumably §2), the custom norm is introduced without an immediate comparison table or remark contrasting it with the standard sup-norm on the dense subspace; adding this would clarify the modification.
The preservation of the slice diameter two property under the quotient map is stated via a direct estimate; citing the precise inequality or lemma number used for the diameter bound would make the argument easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript, which accurately summarizes the constructions and results. We appreciate the recommendation for minor revision. No specific major comments or required modifications were detailed in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit constructions and direct verifications

full rationale

The manuscript supplies explicit constructions: a concrete non-complete normed space (built as a suitable dense subspace equipped with a custom norm) that satisfies both the Daugavet property (verified via the equivalent formulation using rank-one operators) and Gâteaux differentiability at every nonzero vector (verified by direct computation of the directional derivative limit). The dual-norm observation follows from a short general argument using the Daugavet property to produce norm-attaining functionals that prevent differentiability. For the quasilacunary Müntz subspaces, the paper recalls the standard Müntz theorem criterion for isomorphism to c0, then shows the quotient by the subspace fails Daugavet by exhibiting a rank-one operator whose norm is strictly less than 1 + ||T||, while the slice-diameter-two property is preserved by a direct slice-diameter estimate that does not rely on completeness. No step reduces by definition, fitting, or self-citation chain to its own inputs; all load-bearing claims rest on external definitions and direct computations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard definitions from functional analysis; no free parameters, invented entities, or ad-hoc axioms beyond domain conventions.

axioms (2)

standard math Gâteaux differentiability of norms and the Daugavet property as standard notions in normed spaces
Invoked throughout as background from the field.
domain assumption Müntz spaces and C[0,1] having the listed isomorphism and quotient properties
Assumes these classes behave as described in prior literature.

pith-pipeline@v0.9.0 · 5395 in / 1445 out tokens · 154999 ms · 2026-05-07T17:27:58.464105+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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