Epsilon-Isometries in Banach Spaces

• Epsilon-isometries are maps between Banach spaces that preserve distances within a fixed epsilon error, providing an approximate metric structure.
• The theory examines conditions under which these nonlinear approximations can be linearly recovered, with cardinality injectivity ensuring universal left-stability.
• Under separability and ℓ1-exclusion, epsilon-isometries can yield exact isometric embeddings into the bidual, facilitating the transfer of convexity and smoothness properties.

An $\varepsilon$-isometry is a map between Banach spaces that preserves distances up to a uniform additive error. In the standard formulation, if $X$ and $Y$ are real Banach spaces and $\varepsilon\ge 0$, a map $f:X\to Y$ is an $\varepsilon$-isometry when

$$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$

and it is called standard when $f(0)=0$ (Bao et al., 2013, Dai et al., 2013). The modern theory studies when such nonlinear approximate isometries can be linearly recovered, when they force exact linear isometric embeddings, and which Banach spaces admit uniform stability results for all targets. Two 2013 papers develop the structural core of this theory: one characterizes universal left-stability in terms of cardinality injectivity (Bao et al., 2013), while the other shows that, under separability and $\ell_1$-exclusion hypotheses, an $\varepsilon$-isometry already yields a genuine linear isometry into the bidual and transfers fine convexity and smoothness properties (Dai et al., 2013).

1. Definitions, normalization, and the stability problem

For a standard $X$0-isometry $X$1, the closed linear span of its range is denoted

$X$2

A pair $X$3 is called stable if there exists $X$4 such that for every $X$5 and every standard $X$6-isometry $X$7, there is a bounded linear operator

$X$8

satisfying

$X$9

In this form, stability means that an approximate isometric embedding of $Y$0 into $Y$1 can be linearly recovered from its image up to an error controlled uniformly by $Y$2 (Bao et al., 2013).

The theory is motivated by several exact and approximate rigidity results. Mazur–Ulam asserts that every surjective isometry between Banach spaces is affine. Figiel’s theorem shows that if $Y$3 is a standard isometry, then there exists a linear operator $Y$4 with $Y$5 and $Y$6 for all $Y$7. Omladič–Šemrl proved that for surjective $Y$8-isometries one obtains the sharp estimate

$Y$9

for some surjective linear isometry $\varepsilon\ge 0$0. For nonsurjective approximate isometries, however, Qian asked whether one can always recover a bounded linear operator $\varepsilon\ge 0$1 with

$\varepsilon\ge 0$2

and this is false in general; the failure is linked to uncomplemented subspaces (Bao et al., 2013).

Two global notions organize the subject. A Banach space $\varepsilon\ge 0$3 is universally left-stable if $\varepsilon\ge 0$4 is stable for every Banach space $\varepsilon\ge 0$5. A Banach space $\varepsilon\ge 0$6 is universally right-stable if $\varepsilon\ge 0$7 is stable for every Banach space $\varepsilon\ge 0$8. Earlier work recalled in the literature shows that, up to linear isomorphism, the universally right-stable spaces are exactly the Hilbert spaces, whereas the left-hand theory is substantially richer (Bao et al., 2013).

2. Functional transfer and linear recovery mechanisms

A central operator-theoretic tool is the Cheng–Dong–Zhang estimate. For a standard $\varepsilon\ge 0$9-isometry $f:X\to Y$0 and every $f:X\to Y$1, there exists $f:X\to Y$2 with

$f:X\to Y$3

such that

$f:X\to Y$4

This estimate transfers functionals on $f:X\to Y$5 to controlled functionals on $f:X\to Y$6, and it is the starting point for constructing a bounded linear operator from $f:X\to Y$7 back to $f:X\to Y$8 (Bao et al., 2013, Dai et al., 2013).

The same paper introduces the subspace

$f:X\to Y$9

where

$\varepsilon$0

In one form of the stability theorem, if $\varepsilon$1 is reflexive and this $\varepsilon$2 is complemented, then a stable linear recovery operator exists. This precursor result makes explicit the role of complementability in the nonsurjective theory and clarifies why Qian’s counterexample is not accidental but structural (Bao et al., 2013).

The functional estimate also underlies later exact-embedding theorems. Its significance is that approximate distance preservation gives uniform dual control even when $\varepsilon$3 is nonlinear and not onto. This suggests that the effective obstruction to stability is not the nonlinearity of $\varepsilon$4 alone but the ambient linear structure of the codomain and the complementability of the image span. That implication is made precise in the universal theory (Bao et al., 2013).

3. Universal left-stability and cardinality injective spaces

The main structural characterization is expressed in terms of cardinality injectivity. A Banach space $\varepsilon$5 is cardinality injective if there exists $\varepsilon$6 such that whenever $\varepsilon$7 is isometrically embedded into another Banach space $\varepsilon$8 with

$\varepsilon$9

there is a projection $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$0 with

$$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$1

The paper notes that this is equivalent to an extension property for bounded operators defined on subspaces of spaces of cardinality at most $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$2 (Bao et al., 2013).

Theorem 3.3 gives the exact classification: for a Banach space $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$3, the following are equivalent.

1. There exists $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$4 such that for every Banach space $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$5, every $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$6, and every standard $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$7-isometry $$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$8, there is a bounded linear operator

$$\bigl|\,\|f(x)-f(y)\|-\|x-y\|\,\bigr|\le \varepsilon \qquad \text{for all }x,y\in X,$$9

satisfying

$f(0)=0$0

2. $f(0)=0$1 is a cardinality injective space (Bao et al., 2013).

The proof of sufficiency uses the functional estimate (1.3) to build a linear map $f(0)=0$2 from a suitable family of coordinate functionals. If $f(0)=0$3 is the closed span of $f(0)=0$4, then $f(0)=0$5, so cardinality injectivity yields a bounded projection $f(0)=0$6. The operator $f(0)=0$7 then satisfies the desired estimate, with the paper using a constant of the form $f(0)=0$8, where $f(0)=0$9 is the projection constant. The converse uses a lemma showing that if $\ell_1$0 sits uncomplemented in a space $\ell_1$1 of the same cardinality, then one can construct a standard $\ell_1$2-isometry $\ell_1$3 with $\ell_1$4 for which no bounded linear recovery operator exists (Bao et al., 2013).

Several corollaries sharpen the picture. Universal left-stability is invariant under linear isomorphism. Moreover, if for each Banach space $\ell_1$5, each $\ell_1$6, and each standard $\ell_1$7-isometry $\ell_1$8, there exists some constant $\ell_1$9 and a bounded linear operator $\varepsilon$0 with

$\varepsilon$1

then $\varepsilon$2 is in fact universally left-stable. Thus targetwise existence of a stability estimate upgrades to uniform universality (Bao et al., 2013).

For dual spaces the result becomes more rigid. Theorem 2.7 states that if $\varepsilon$3 is a universally left-stable Banach space, then there exists an injective conjugate space $\varepsilon$4 such that

$\varepsilon$5

Corollary 2.8 then shows that if $\varepsilon$6 is a universally left-stable dual Banach space, then $\varepsilon$7 is injective. Combined with the fact that every injective Banach space is universally left-stable, this yields

$\varepsilon$8

for dual Banach spaces. The paper further identifies such spaces as complemented $\varepsilon$9-closed subspaces of $X$00, aligning the approximate-isometry theory with the classical Goodner–Kelley–Nachbin description of injective spaces (Bao et al., 2013).

4. Exact linear isometric embeddings from nonlinear $X$01-isometries

A complementary line of work asks when a nonlinear $X$02-isometry forces the existence of an exact linear isometric embedding. Theorem 3.3 of the second 2013 paper states that if

• $X$03 is separable,
• $X$04 is a Banach space containing no closed subspace isomorphic to $X$05,
• $X$06 is an $X$07-isometry with $X$08,

then there exists an isometry

$X$09

This is an exact isometric embedding into the bidual, obtained from a nonlinear approximate isometry (Dai et al., 2013).

Corollary 3.4 identifies two important codomain classes in which the bidual conclusion descends to the original space. If $X$10 is either the James space $X$11 or a reflexive Banach space, then there exists a linear isometry

$X$12

The reflexive case is immediate from $X$13, while the James space case uses that $X$14 is isometric to $X$15 (Dai et al., 2013).

The proof combines Rosenthal’s $X$16-theorem with the dual-functional estimate described above. One takes a dense sequence $X$17 in $X$18, extracts subsequences so that $X$19 is weak$X$20-Cauchy in $X$21, and by a diagonal argument defines

$X$22

Using the Cheng–Dong–Zhang theorem, for each $X$23 one finds $X$24 with

$X$25

which yields

$X$26

Hence $X$27 is an isometry on the dense set $X$28 and extends uniquely to all of $X$29 (Dai et al., 2013).

This result is presented as an $X$30-version of the Godefroy–Kalton theorem. The exact Godefroy–Kalton theorem says that if $X$31 is separable and there exists an exact isometry $X$32, then $X$33 contains a linear isometric copy of $X$34. The $X$35-theorem shows that, under a no-$X$36 hypothesis on $X$37, approximate nonlinear isometries still encode exact linear geometry (Dai et al., 2013).

5. Set-valued formulations and transfer of convexity and smoothness

The same paper develops a set-valued mapping formulation of the stability problem. For $X$38, with $X$39, Problem 4.1 asks whether one can select, continuously and linearly, functionals on $X$40 approximating each $X$41. For $X$42, the paper defines

$X$43

Lemma 4.2 shows that $X$44 is convex and weak$X$45-usco at each point of $X$46, that it contains a minimal convex norm–weak$X$47-usco mapping, and that if $X$48 is separable then $X$49 admits a selection that is norm–weak$X$50 continuous on a norm-dense $X$51 subset of $X$52 (Dai et al., 2013).

These selection results are then used to transfer geometric properties. Proposition 4.3 states that if there exists a norm–weak$X$53 continuous selection of

$X$54

then $X$55 is smooth. In particular, if $X$56 is rotund, then $X$57 is rotund, hence $X$58 is smooth (Dai et al., 2013).

Proposition 4.5 provides a systematic dictionary between regularity properties of $X$59 and those of $X$60. If $X$61 is an $X$62-isometry with $X$63, then:

1. if $X$64 is smooth, then $X$65 is rotund;
2. if $X$66 is uniformly Gateaux smooth, then $X$67 is weakly uniformly rotund;
3. if $X$68 is Fréchet smooth, then $X$69 is strongly rotund;
4. if $X$70 is strongly rotund, then $X$71 is Fréchet smooth;
5. if $X$72 is uniformly smooth, then $X$73 is uniformly rotund;
6. if $X$74 is uniformly rotund, then $X$75 is uniformly smooth (Dai et al., 2013).

The mechanism is again dual-functional control. The authors define a map $X$76, where $X$77 is the span of the selected functionals, by taking a weak$X$78 limit of normalized vectors $X$79. This $X$80 is an isometry and satisfies

$X$81

Through this intertwining relation and the duality statements summarized in Proposition 2.3 of the paper, fine smoothness and rotundity pass from $X$82 to $X$83 (Dai et al., 2013).

6. Related notions, neighboring usages, and common distinctions

The term “isometry” appears in several adjacent literatures, but these notions are not interchangeable.

In compressed sensing, the restricted isometry property is a property of a matrix rather than a nonlinear map between Banach spaces. A matrix $X$84 satisfies the $X$85-restricted isometry property if

$X$86

for every $X$87-sparse vector $X$88. A 2014 paper gives a conditional deterministic construction of such matrices from quadratic residues, obtaining sparsity

$X$89

for the Paley matrix under a discrepancy conjecture for the Legendre symbol (Bandeira et al., 2014). Despite the shared word “isometry,” this is a matrix-analytic near-orthogonality condition, not the Banach-space notion of an $X$90-isometry.

In point-configuration matching, the relevant approximation is often multiplicative rather than additive. An announcement on the Orthogonal Procrustes problem and $X$91-diffeomorphisms uses conditions of the form

$X$92

and also the additive estimate

$X$93

with Kabsch’s algorithm providing the optimal rigid motion once correspondences are known (Charalambides et al., 2017). This is a finite-dimensional approximation theory of congruence, not the stability theory of nonlinear maps between Banach spaces.

A different neighboring rigidity phenomenon arises for exact isometries with bounded displacement. For metric spaces admitting a transitive action by an exponential solvable Lie group, every bounded isometry is a Clifford–Wolf isometry, equivalently an isometry of constant displacement, and such isometries are precisely the center of the transitive solvable group (Wolf, 2015). This literature studies how weak global bounds on displacement force exact homogeneous structure; it does not define $X$94-isometries, but it is closely related in spirit to the theme that approximate symmetry can collapse to rigid exact symmetry.

These distinctions suggest a useful taxonomy. In Banach-space theory, an $X$95-isometry is an additive distance-preserving approximation for a map $X$96. In compressed sensing, “restricted isometry” refers to almost-Euclidean behavior on sparse vectors. In point-cloud matching, $X$97-distortion is typically multiplicative. In homogeneous geometry, bounded displacement is a rigidity condition for exact isometries. The common thread is approximate preservation of metric structure, but the ambient categories, recovery problems, and structural conclusions differ substantially (Bao et al., 2013, Dai et al., 2013, Bandeira et al., 2014, Charalambides et al., 2017, Wolf, 2015).