Dilworth–Tabor Theorem: Stability of Approximate Isometries

• The Dilworth–Tabor Theorem is a foundational result in metric geometry and stability theory, defining how approximate isometries in Banach spaces guarantee proximity to true surjective isometries.
• It provides explicit quantitative bounds, showing that every nearly surjective d-isometry is within a controllable sup-norm distance of an exact linear isometry.
• The theorem unifies metric and algebraic perspectives using a Cantor–Bernstein reduction and connects classic concepts like the Mazur–Ulam theorem and Gromov–Hausdorff distance.

The Dilworth–Tabor Theorem is a foundational result in the stability theory of functional equations and metric geometry, generalizing the Hyers–Ulam stability of isometric mappings. It asserts that every approximately surjective approximate isometry between Banach spaces is close, in the sup-norm, to a genuine surjective linear isometry. The theorem provides sharp quantitative bounds on the proximity, connects metric and algebraic structures, and has been shown to be equivalent to a variety of deep statements regarding isometries, Gromov–Hausdorff distance, and the structure of Banach spaces. Recent developments reduce the theorem to isometric stability results via a metric Cantor–Bernstein argument and clarify its reach within modern analysis.

1. Formal Statement and Quantitative Bounds

Let $V$ and $W$ be real Banach spaces, and consider a mapping $f : V \to W$ satisfying three conditions: (1) $f(0) = 0$, (2) $f$ is a $d$-isometry, i.e., $|\|f(x)-f(x')\|-\|x-x'\|| \le d$ for all $x,x' \in V$, and (3) $f$ maps $V$ $\delta$-surjectively onto a closed linear subspace $L \subset W$—that is, the Hausdorff distance between $f(V)$ and $L$ is at most $\delta$. The Dilworth–Tabor theorem asserts the existence of a bijective linear isometry $U: V \to L$ such that $\|f-U\| \le M$, with explicit choices:

• $M=12d+5$ (Dilworth's original bound);
• $M=2d+35$ if $L=W$ (Tabor's improvement).

This bound was further sharpened to $M=2d+2$ generally, and $M=2d$ for surjective mappings, by Šemrl–Väisälä, and their result is optimal. The theorem fundamentally quantifies and unifies the notions of approximate isometries and surjectivity in Banach spaces (Bogatyi et al., 28 Dec 2025).

2. Definitions and Preliminaries

Key concepts in the statement and analysis of the Dilworth–Tabor theorem include:

• $d$-isometry: A map $f : V \to W$ between normed spaces is a $d$-isometry if

$$|\|f(x)-f(x')\| - \|x-x'\|| \le d \quad \forall x,x'\in V.$$

• $\delta$-surjectivity: For metric spaces $X, Y$, a mapping $f : X \rightarrow Y$ is $\delta$-surjective if the Hausdorff distance $d_H(f(X), Y) \le \delta$.
• Cardinality-homogeneous spaces: A metric space $X$ where every nonempty open subset has the same cardinality as $X$; the theorem's sharpness and generalizations often assume this structural richness.
• Gromov–Hausdorff distance: For compact metric spaces, the infimum of distortions over all correspondences; for Banach spaces, two spaces have finite Gromov–Hausdorff distance iff they are linearly isomorphic up to arbitrary precision.

These definitions enable a metric and topological framework that extends the classical isometry problem to more general settings (Bogatyi et al., 28 Dec 2025).

3. Proof Structure and Reductions

The modern approach to the Dilworth–Tabor theorem proceeds via a two-step reduction:

1. From an approximately surjective approximate isometry $f$, construct (using the Cantor–Bernstein argument) a genuine bijective approximate isometry $\tilde f$ between equicardinal cardinality-homogeneous metric spaces. Specifically, given $f: X \to Y$ ($\delta$-surjective $d$-isometry), one produces a bijection satisfying

$$|\|\tilde f(x) - \tilde f(x')\| - \|x-x'\|| \le d + 2.$$

2. Apply the Gevirtz–Omladič–Šemrl theorem: any surjective $D$-isometry $F: V \to W$ is within $2D$ in sup-norm of a true linear isometry $U: V \to W$: $\|F - U\| \le 2D$.

Taking $D = d+2$, the combined error bound is $2d+4$ (plus an arbitrarily small $\delta$-error). This reduction encapsulates the metric and algebraic content, providing canonical constants and demonstrating the absence of the need for further Banach-space geometric lemmas once the metric-space bijection is constructed (Bogatyi et al., 28 Dec 2025).

4. Historical Context and Relationship to Classical Results

The theorem traces its origins to the Hyers–Ulam stability theory of mappings and to S.J. Dilworth and J. Tabor, who formulated the stability of approximate isometries for Banach spaces in the 1970s and 1980s. The claim that any mapping which is close to surjective and close to being an isometry is itself close to an actual surjective isometry is a global generalization of the Mazur–Ulam theorem, which states that all surjective isometries between real Banach spaces are affine. The reduction to Gevirtz (1983) and Omladič–Šemrl (1995) places the result within the framework of sharp quantitative stability for functional equations.

Recent work clarifies that for cardinality homogeneous spaces, several topologies (injective, surjective, bijective Gromov–Hausdorff distances) coincide and are characterized as

$$d_{GH}(X, Y) = \frac{1}{2}\inf\{\mathrm{dis}(f) : f \text{ a bijection } X \to Y\}.$$

Thus, the Mazur–Ulam theorem can be restated and proven via Gromov–Hausdorff continuity (Bogatyi et al., 28 Dec 2025).

5. Metric–Geometric Lemmas and Cantor–Bernstein Reduction

A central component of the proof is a variant of the Cantor–Bernstein argument for metric spaces, which enables the construction of a bijection from two approximately inverse (possibly non-linear) maps while controlling their distortion. If $f: X \to Y$ ($\delta$-surjective $d$-isometry), and $g: Y \to X$ is a right-inverse up to error, one can construct a bijection $\tilde f: X \to Y$ such that

$|\|\tilde f(x) - \tilde f(x')\|-\|x-x'\|| \le d+2,$

where $\tilde f$ is a $(d+2)$-isometry. This reduction critically depends on the cardinality homogeneity of the underlying spaces and the large set-theoretical symmetries it provides (Bogatyi et al., 28 Dec 2025). The lemma formalizes and extends classical tricks to handle surjective stability in the metric category.

6. Significance and Implications

The Dilworth–Tabor theorem provides a complete quantitative solution for the approximate-surjection problem in the category of Banach spaces (and, more generally, cardinality-homogeneous metric spaces). It reveals that the existence of a $\delta$-surjective approximate isometry implies the presence of a genuine surjective isometry within bounded sup-norm distance, characterized by sharp constants. This framework unifies metric geometry and linear structure, establishes when Banach spaces are linearly isomorphic up to arbitrary small perturbation, and gives a transparent metric criterion for isometry based on Gromov–Hausdorff distance. The result’s reduction to known surjective isometry stability problems highlights the sufficiency of set-theoretic and metric-geometric tools, eliminating the need for new Banach-space-specific geometric analysis beyond this context (Bogatyi et al., 28 Dec 2025).

A plausible implication is that the theorem, as formulated for equicardinal cardinality-homogeneous spaces, may not directly extend with sharp constants to spaces lacking this structural property. This suggests an inherent limitation tied to the symmetry and set-theoretical richness of such spaces.

7. Connections with Related Theorems

The Dilworth–Tabor theorem is conceptually adjacent to the Gevirtz–Omladič–Šemrl theorem on surjective stability, the Mazur–Ulam theorem, and the general theory of Gromov–Hausdorff distances for metric and Banach spaces. Its sharp reductions reveal the equivalence of several natural metric and algebraic proximity notions in the Banach category, with corollaries that clarify the isometric and topological structure of infinite-dimensional normed spaces. Its methodology and consequences are also relevant to ongoing research in the approximation and rigidity of mappings between infinite-dimensional spaces (Bogatyi et al., 28 Dec 2025).