TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Bakushinskii veto for discontinuous pseudoinverses

Statement

Assume Countable Choice. Let $K : X \to Y$ be a compact linear operator between real or complex Hilbert spaces over one scalar field, with Hilbert-space Moore–Penrose inverse $K^{†} : D (K^{†}) \to X$ . Suppose $K^{†}$ is discontinuous on $D (K^{†})$ with its norm inherited from $Y$ . Let $(R_{α})_{α > 0}$ be maps $Y \to X$ such that $R_{α} y \to K^{†} y$ for every $y \in D (K^{†})$ as $α ↓ 0$ , and let $α : Y \to (0, \infty)$ be a deterministic noise-level-free rule. For each exact datum $y \in D (K^{†})$ and $δ > 0$ , set $E_{y} (δ) : = \sup {∥ R_{α (z)} z - K^{†} y ∥_{X} : z \in Y, ∥ z - y ∥_{Y} \leq δ}$ . If $E_{y} (δ) \to 0$ as $δ ↓ 0$ for every $y \in D (K^{†})$ , then $K^{†}$ is continuous on $D (K^{†}) .$ Thus, when $K^{†}$ is discontinuous, no such noise-level-free rule yields worst-case convergence over each exact datum's full admissible error balls. The limit is required separately for each exact datum, not uniformly over $y$ . This veto does not exclude convergence on restricted solution or error classes, stochastic noise models, or finite-dimensional problems.

Statement references

Compact Hilbert inverse model

Hilbert-space Moore–Penrose inverse and its domain

Noise-level-free parameter-choice rule

Clason, Regularization of Inverse Problems, Theorem 4.8

Sherry, Inverse Problems: Lecture Notes 2024, Theorem 3.12

Facts & Assumptions

Given: Countable Choice, a compact linear map $K : X \to Y$ between real or complex Hilbert spaces over one scalar field, its Moore–Penrose inverse $K^{†}$ discontinuous on $D (K^{†})$ , a family $(R_{α})_{α > 0}$ that converges pointwise to $K^{†}$ on this domain, a noise-level-free rule $α : Y \to (0, \infty)$ , and the convergence condition $E_{y} (δ) \to 0$ for each $y \in D (K^{†})$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): For every family $(X_{n})_{n \in N}$ of nonempty sets indexed by $N$ , there is a function $f$ with domain $N$ such that $f (n) \in X_{n}$ for every $n$ .

[F1]

Compact linear inverse model between Hilbert spaces: The compact inverse model has real or complex Hilbert parameter and data spaces over one scalar field and a compact linear forward map $K : X \to Y$ .

[F2]

Hilbert-space Moore–Penrose inverse: The inverse has domain $D (K^{†}) = R (K) \oplus R (K)^{⊥}$ and maps $r + z$ to the unique minimum-norm solution of $K x = r$ .

[F3]

Noise-level-free parameter-choice rule: For the fixed operator and reconstruction family, $α$ is a function of the observed datum alone; its value does not take the exact datum or noise bound as input.

Proof

1.1givenA1F1F2F3algebra

Fix $y \in D (K^{†})$ and put $T (y) : = R_{α (y)} y$ . By [F3], this vector is independent of $δ$ . Since $y$ itself satisfies $∥ y - y ∥_{Y} = 0 \leq δ$ for every $δ > 0$ , the supremum defining $E_{y} (δ)$ gives $∥ T (y) - K^{†} y ∥_{X} \leq E_{y} (δ)$ for every $δ > 0$ . Letting $δ ↓ 0$ proves $T (y) = K^{†} y$ .

2.1givenstep 1.1F2F3choosealgebra∎

Fix $y \in D (K^{†})$ and $ε > 0$ . By the assumed limit choose $r > 0$ such that $E_{y} (r) < ε$ . For any $z \in D (K^{†})$ with $∥ z - y ∥_{Y} \leq r$ , step 1.1 applied to $z$ gives $R_{α (z)} z = K^{†} z$ , while $z$ belongs to the ball in $E_{y} (r)$ . Hence $∥ K^{†} z - K^{†} y ∥_{X} \leq E_{y} (r) < ε$ . This is continuity of $K^{†}$ at the arbitrary point $y$ in its subspace domain, contradicting the hypothesis.

Depends on

Used by

Conditional stability on a power-source class Example

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christian Clason, Regularization of Inverse Problems: Lecture Notes, Theorem 4.8 (standard reference, not scraped)
Ferdia Sherry, Inverse Problems: Lecture Notes 2024, Theorem 3.12 (standard reference, not scraped)