LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Index source elements are minimum-norm solutions

Statement

Assume Countable Choice. Let $K : X \to Y$ be compact between real or complex Hilbert spaces over the same scalar field, let $ψ$ be a bounded nonnegative spectral index with $ψ (0) = 0$ , and let $x = ψ (∣ K ∣) w$ for some $w \in X$ . Then $x \in N (K)^{⊥}$ ; consequently, among all $z \in X$ satisfying $K z = K x$ , $x$ is the unique vector of minimum norm.

Statement references

Compact Hilbert-space inverse model

Clason, Regularization of Inverse Problems, Chapter 3

Hofmann and Mathé, A Note on the Modulus of Continuity, Section 1

Facts & Assumptions

Given: Countable Choice, compact $K : X \to Y$ between real or complex Hilbert spaces, an admissible index function $ψ$ with $ψ (0) = 0$ , and $x = ψ (∣ K ∣) w$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed by the compact singular-value decomposition supplier.

[F1]

Singular value decomposition for compact operators: A compact operator has orthonormal right singular vectors spanning $N (K)^{⊥}$ , with $∣ K ∣ v_{j} = s_{j} v_{j}$ for each positive singular value $s_{j}$ , and $X = N (K)^{⊥} \oplus N (K)$ .

[F2]

Spectral-index source balls for compact operators and Compact spectral filters, source conditions, and qualification: A bounded spectral index acts by $ψ (s_{j})$ on each right singular vector and by $ψ (0) = 0$ on the kernel.

[F3]

Hilbert space, Real and complex inner-product spaces and their induced length, and Orthogonality and the orthogonal complement: The Hilbert norm is induced by the inner product; if $x ⊥ n$ , then $∥ x + n ∥^{2} = ∥ x ∥^{2} + ∥ n ∥^{2}$ .

Proof

1.1givenA1F1F2construct

By [F1], write $w = w_{0} + \sum_{j} w_{j} v_{j}$ in norm, with $w_{0} \in N (K)$ and the $v_{j}$ an orthonormal basis of $N (K)^{⊥}$ . The compact spectral multiplier in [F2] gives $x = ψ (∣ K ∣) w = ψ (0) w_{0} + \sum_{j} ψ (s_{j}) w_{j} v_{j} = \sum_{j} ψ (s_{j}) w_{j} v_{j}$ . This norm-convergent sum is a limit of finite linear combinations of the $v_{j}$ , so [F1] places $x$ in their closed span $N (K)^{⊥}$ ; if the positive singular set is empty, the sum and closed span are both zero. Thus $x ⊥ N (K)$ .

2.1givenstep 1.1F3algebra∎

If $K z = K x$ , then $n : = z - x \in N (K)$ . By step 1.1, $x ⊥ n$ , so [F3] gives $∥ z ∥^{2} = ∥ x + n ∥^{2} = ∥ x ∥^{2} + ∥ n ∥^{2} \geq ∥ x ∥^{2}$ , with equality only when $n = 0$ , that is, $z = x$ . Thus $x$ is the unique minimum-norm solution.

Depends on

Used by

Power source ball Definition
Power source balls are weighted statistical ellipsoids Lemma
Conditional modulus on a power source ball Theorem
Deterministic minimax lower bound on power source balls Theorem

Dependency tree · two levels

46 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 — Chapter 3, compact SVD and minimum-norm solutions (standard reference, not scraped)
Bernd Hofmann and Peter Mathé, A Note on the Modulus of Continuity for Ill-Posed Problems in Hilbert Space (standard reference, not scraped)