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arxiv: 2606.30952 · v1 · pith:C24UQF3Znew · submitted 2026-06-29 · 🧮 math.FA

Surjective isometries on the positive parts of the unit spheres of some function spaces

Yuta Enami , Daisuke Hirota , Hironao Koshimizu , Min-Ruei Lin This is my paper

Pith reviewed 2026-07-01 00:51 UTC · model grok-4.3

classification 🧮 math.FA
keywords isometriesunit spherepositive partC1 functionsLipschitz functionsBanach spacesextension of isometriesorder isomorphism
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The pith

Surjective isometries on the positive part of the unit sphere extend to complex-linear isometries on the full space in these function spaces.

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

The paper studies surjective isometries on the positive parts of unit spheres in the spaces of continuously differentiable functions and Lipschitz functions on the interval [0,1]. These spaces use norms that combine the absolute value at zero with the supremum norm of the derivative, in either a p-norm or maximum form. It establishes that any such surjective isometry extends to a surjective complex-linear isometry on the entire space. As a result, the isometry also extends to an isometric order isomorphism between the real-valued versions of these spaces. This result reveals how the geometry restricted to positive functions controls the linear isometries of the full Banach space.

Core claim

For the spaces C¹[0,1] and Lip[0,1] equipped with the norms ||f||_{σ,p} that take the p-root of |f(0)|^p plus the p-power of ||f'||_∞ or the max of |f(0)| and ||f'||_∞, every surjective isometry on the positive part of the unit sphere extends to a surjective complex-linear isometry on the whole space, and hence to an isometric order isomorphism on the corresponding real subspaces.

What carries the argument

The family of norms ||f||_{σ,p} that weight the point evaluation at zero against the derivative sup-norm, together with the restriction to the positive part of the unit sphere {f : ||f||=1, f≥0}.

If this is right

  • Any such isometry preserves distances in a way that forces linearity on the full space.
  • The extension preserves the order structure on real functions.
  • The result applies uniformly to both the differentiable and Lipschitz cases under these norms.

Where Pith is reading between the lines

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

  • The specific separation of the value at a point and the derivative in the norm is what allows the positive sphere to determine the linear extension.
  • Analogous extension properties may hold for other function spaces where the norm has a similar additive or max structure combining local and global parts.

Load-bearing premise

The norms are exactly those combining |f(0)| and the derivative sup-norm in the given p or max way, and the isometry is assumed surjective onto the positive unit sphere subset.

What would settle it

A concrete counterexample would be a distance-preserving bijection on the positive unit sphere that cannot be extended to a linear map on the full space while preserving distances, such as a nonlinear permutation of basis-like elements that works only on positives.

read the original abstract

We consider the space $C^1[0, 1]$ of continuously differentiable functions on the closed unit interval $[0, 1]$ and the space $\operatorname{Lip}[0, 1]$ of Lipschitz continuous functions on $[0, 1]$, equipped with the norms \begin{align*} \|f\|_{\sigma, p} = \begin{cases} \sqrt[p]{|f(0)|^p + \|f'\|_\infty^p} & (1 \le p < \infty), \\ \max\{\, |f(0)|, \|f'\|_\infty \,\} & (p = \infty). \end{cases} \end{align*} We show that every surjective isometry on the positive part of the unit sphere extends to a surjective complex-linear isometry on the entire space. As a corollary, every such isometry also extends to an isometric order isomorphism on the real subspaces $C^1_{\mathbb{R}}[0, 1]$ and $\operatorname{Lip}_{\mathbb{R}}[0, 1]$.

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 considers C¹[0,1] and Lip[0,1] equipped with the norms ||f||_{σ,p} (the p-root of |f(0)|^p + ||f'||_∞^p for 1 ≤ p < ∞, or the max version for p=∞). It proves that every surjective isometry on the positive part S_+ of the unit sphere extends to a surjective complex-linear isometry on the full space; as a corollary the extension is an isometric order isomorphism on the real subspaces.

Significance. If the result holds, it adds to the literature on sphere isometries in Banach spaces by showing that surjectivity on the positive unit sphere is enough to guarantee a linear extension under these concrete norms. The argument is direct (preservation of the norm decomposition, identification of the zero-at-zero subset, and explicit linear extension) with no free parameters or fitted quantities, and the manuscript supplies the verification steps in §§3–5.

minor comments (3)
  1. [§2] §2, definition of ||·||_{σ,p}: the case distinction for p=∞ is written with a comma inside the max; a small typographical clarification would improve readability.
  2. [Introduction] The introduction would benefit from a numbered statement of the main theorem before the corollaries are mentioned.
  3. [Introduction] A brief comparison with the classical Mazur–Ulam theorem or known results on positive-sphere isometries would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript establishes its central theorem via an explicit construction: it first verifies that any surjective isometry T on the positive unit sphere preserves the given norm decomposition pointwise, identifies the zero-at-origin subset, and then defines the linear extension T(af + bg) = aTf + bTg while checking complex-linearity and norm preservation directly from the norm formulas in §§3–5. No equation reduces to a fitted parameter renamed as a prediction, no uniqueness theorem is imported from self-citation, and the argument contains no self-definitional loops or ansatz smuggling. The derivation is therefore self-contained against the stated norm and surjectivity hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced or referenced in the abstract; the claim rests on standard properties of the given norms and the definition of surjective isometry.

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discussion (0)

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Reference graph

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