Non-Archimedean Tarski-Maligranda Inequalities
Pith reviewed 2026-05-15 17:31 UTC · model grok-4.3
The pith
Non-Archimedean normed spaces satisfy distinct versions of the Tarski-Maligranda inequalities.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We derive non-Archimedean versions of Tarski-Maligranda inequalities. In spaces where the norm satisfies the ultrametric inequality, the relations between norms of sums, differences, and absolute values take forms that contrast with the Archimedean case.
What carries the argument
The ultrametric inequality ||x+y|| ≤ max{||x||, ||y||} in a linear space over a valued field, which replaces the triangle inequality and drives the modified bounds.
If this is right
- The inequalities apply to any non-Archimedean normed linear space without additional assumptions beyond the ultrametric property.
- Expressions involving |r| - |s| and |r - s| + |r + s| simplify differently under the max operation.
- Classical bounds may not hold, leading to stricter or altered estimates in these spaces.
- Derivations can be carried out directly from the definition of the non-Archimedean norm.
Where Pith is reading between the lines
- These results indicate that inequality theory in analysis must be developed separately for Archimedean and non-Archimedean cases.
- Similar adaptations might be needed for other classical inequalities when moving to valued fields.
- Computational verification in specific examples like the p-adic numbers could confirm the bounds quickly.
Load-bearing premise
The norm on the space must satisfy the ultrametric inequality instead of the ordinary triangle inequality.
What would settle it
Finding a linear space over a valued field with an ultrametric norm where one of the derived inequalities fails to hold for some vectors.
read the original abstract
In 1930, Tarski observed that \begin{align*} \bigg||r|-|s|\bigg|=|r-s|+ |r+s|-(|r|+|s|), \quad \forall r, s \in \mathbb{R}. \end{align*} In 2008, Maligranda converted the previous equality into inequalities that are valid in every normed linear space. We derive non-Archimedean versions of Tarski-Maligranda inequalities. Difference between Archimedean and non-Archimedean inequalities is surprising.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript derives non-Archimedean analogues of Tarski's 1930 equality and Maligranda's 2008 inequalities for normed linear spaces over valued fields, replacing the usual triangle inequality with the ultrametric inequality ||x+y|| ≤ max{||x||,||y||}. The abstract states that the resulting inequalities differ from their Archimedean counterparts in a surprising way.
Significance. If the derivations are correct, the work supplies explicit non-Archimedean versions of classical norm inequalities that follow directly from the ultrametric property and the vector-space axioms. This constitutes a parameter-free extension that may be useful in p-adic analysis and non-Archimedean functional analysis; the direct derivation from the defining inequality is a methodological strength.
minor comments (2)
- [Abstract] The abstract asserts that the difference between Archimedean and non-Archimedean inequalities is 'surprising' but does not indicate the concrete form of that difference; a single illustrative comparison in the abstract or introduction would improve readability.
- [§2] Notation for the valued field and the non-Archimedean norm should be introduced once at the beginning of §2 and used consistently thereafter; several passages appear to switch between |·| and ||·|| without explicit remark.
Simulated Author's Rebuttal
We thank the referee for the positive summary, significance assessment, and recommendation for minor revision. No specific major comments were raised in the report.
Circularity Check
No significant circularity; derivation is self-contained from ultrametric definition
full rationale
The paper begins with Tarski's 1930 equality on the reals and Maligranda's 2008 conversion to inequalities valid in any normed linear space, then derives the non-Archimedean analogues by substituting the ultrametric inequality ||x+y|| ≤ max{||x||, ||y||} together with the linear-space axioms over a valued field. These are the standard defining properties of the setting addressed; the claimed inequalities follow directly from them without fitted parameters, self-referential definitions, or load-bearing self-citations. No step reduces by construction to its own inputs, and external references to Tarski and Maligranda supply independent starting points rather than circular justification.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption A non-Archimedean norm satisfies ||x+y|| ≤ max{||x||, ||y||} for all vectors x, y.
discussion (0)
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