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

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Tangential cone control compares derivative and data residuals

Statement

Let $X, Y$ be real Hilbert spaces, let $U \subseteq X$ be open, and let $F : U \to Y$ be Fréchet differentiable at $x \in U$ . Suppose that for some $δ > 0$ with ${\overline{B}}_{δ} (x) \subseteq U$ and some $0 \leq η < 1$ , the local tangential cone condition at $x$ holds on radius $δ$ with constant $η$ . Then for every $h \in X$ with $∥ h ∥_{X} \leq δ$ ,

(1 - η) ∥ F (x + h) - F (x) ∥_{Y} \leq ∥ F^{'} (x) h ∥_{Y} \leq (1 + η) ∥ F (x + h) - F (x) ∥_{Y} .

Statement references

Local tangential cone condition

Christian Clason, Regularization of Inverse Problems, Theorem 9.8

Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, §2.3

Facts & Assumptions

Given: Real Hilbert spaces $X, Y$ , an open set $U \subseteq X$ , a map $F : U \to Y$ Fréchet differentiable at $x \in U$ , a radius $δ > 0$ with ${\overline{B}}_{δ} (x) \subseteq U$ , a constant $0 \leq η < 1$ for which the local tangential cone condition holds at $x$ , and $h \in X$ with $∥ h ∥_{X} \leq δ$ .

[F1]

The local tangential cone condition gives $∥ F (x + h) - F (x) - F^{'} (x) h ∥_{Y} \leq η ∥ F (x + h) - F (x) ∥_{Y}$ for every $h$ with $∥ h ∥_{X} \leq δ$ (Local tangential cone condition).

[F2]

The induced norm of a real inner-product space satisfies $∥ u + v ∥ \leq ∥ u ∥ + ∥ v ∥$ (The induced length is a norm).

Proof

1.1givenF1F2construct

Put $d : = F (x + h) - F (x)$ and $e : = d - F^{'} (x) h$ . By [F1], $∥ e ∥_{Y} \leq η ∥ d ∥_{Y}$ . Since $F^{'} (x) h = d - e$ , [F2] gives $∥ F^{'} (x) h ∥_{Y} \leq ∥ d ∥_{Y} + ∥ e ∥_{Y} \leq (1 + η) ∥ d ∥_{Y}$ .

2.1F1F2step 1.1algebra∎

Since $d = F^{'} (x) h + e$ , [F2] gives $∥ d ∥_{Y} \leq ∥ F^{'} (x) h ∥_{Y} + ∥ e ∥_{Y} \leq ∥ F^{'} (x) h ∥_{Y} + η ∥ d ∥_{Y}$ by [F1]. Rearranging yields $(1 - η) ∥ d ∥_{Y} \leq ∥ F^{'} (x) h ∥_{Y}$ ; combining this with step 1.1 proves both inequalities.

Depends on

Used by

Local uniqueness of an x₀-minimum-norm solution under TCC Theorem

Dependency tree · two levels

13 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 9.8 (standard reference, not scraped)
Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, §2.3 (standard reference, not scraped)