DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

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x₀-minimum-norm solutions of nonlinear equations

Definition

Let $X, Y$ be real Hilbert spaces, let $U \subseteq X$ be a nonempty open set, let $F : U \to Y$ be a forward map, let $y \in Y$ , and let $x_{0} \in X$ . A point $x^{†} \in U$ is an $x_{0}$ -minimum-norm solution of $F (x) = y$ on $U$ if $F (x^{†}) = y$ and

$∥ x^{†} - x_{0} ∥ \leq ∥ x - x_{0} ∥ for every x \in U with F (x) = y .$

This definition asserts no existence or uniqueness; it describes the minimizing property of a supplied exact solution. The minimum is global over the exact solution set on $U$ , with distance measured from the specified reference point $x_{0}$ .

Definition references

Nonlinear forward maps and exact solution sets

Real Hilbert spaces and their induced norm

Christian Clason, Regularization of Inverse Problems, Theorem 9.9

Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, Proposition 1

Depends on

Used by

Dependency tree · two levels

9 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, Theorem 9.9 (standard reference, not scraped)
Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, Proposition 1 (standard reference, not scraped)