LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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The differential annihilates the tangent kernel at a constrained extremum

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a real Banach space, let $U \subseteq X$ be open, let $I : U \to R$ be Fréchet differentiable at $u \in U$ (Fréchet derivative between Banach spaces), and let $G : U \to R^{m}$ be of class $C^{1}$ (C k map between Banach spaces) with $D G (u)$ surjective. If $u$ is a local minimiser or a local maximiser of $I$ on the level set ${G = G (u)}$ , then $D I (u) h = 0$ for every $h \in \ker D G (u)$ .

Facts & Assumptions

Given: A real Banach space $X$ , open $U \subseteq X$ , a map $I$ differentiable at $u$ with differential $D I (u) \in X^{*}$ , a $C^{1}$ map $G$ with $D G (u)$ surjective, and the assumption that $u$ is a local minimiser or local maximiser of $I$ on the level set ${G = G (u)}$ , meaning that for some radius $δ > 0$ one has $I (u) \leq I (z)$ (respectively $I (u) \geq I (z)$ ) for every $z \in U$ with $∥ z - u ∥ < δ$ and $G (z) = G (u)$ .

[F1]

The tangent space of a regular level set is the kernel of the constraint derivative: for every $h \in \ker D G (u)$ there are $ε > 0$ and a $C^{1}$ curve $γ : (- ε, ε) \to X$ with $γ (0) = u$ , $γ^{'} (0) = h$ and $G (γ (t)) = G (u)$ for all $t$ .

[F2]

Chain sum product and composition rules for Banach derivatives, Fréchet derivative between Banach spaces: the composition $t \mapsto I (γ (t))$ is differentiable at $0$ with derivative $D I (u) γ^{'} (0) = D I (u) h$ .

[F3]

Fermat's interior extremum theorem: if $f$ has a local extremum at a point $c$ interior to its domain and is differentiable at $c$ , then $f^{'} (c) = 0$ : a real function on an open interval that is differentiable at an interior point and has a local minimum or local maximum there has derivative $0$ at that point.

[A1]

The Axiom of Choice: the hypothesis under which the level-set parametrisation of [F1] is available.

Proof

technique · direct

Given: The setting above and a vector $h \in \ker D G (u)$ .

1.1givenA1F1

By [F1] choose $ε > 0$ and a $C^{1}$ curve $γ : (- ε, ε) \to X$ with $γ (0) = u$ , $γ^{'} (0) = h$ and $G (γ (t)) = G (u)$ for every $t$ ; by continuity of $γ$ at $0$ and the strict positive radius $δ$ of the local extremum hypothesis, we may shrink $ε$ so that $∥ γ (t) - u ∥ < δ$ for all $t$ .

2.1step 1.1F2

The function $φ (t) : = I (γ (t))$ is defined on the open interval $(- ε, ε)$ , is differentiable at $0$ with $φ^{'} (0) = D I (u) h$ by [F2], and has a local minimum (respectively local maximum) at $t = 0$ : for $∣ t ∣ < ε$ the curve lies in the level set and within distance $δ$ of $u$ , so $φ (0) = I (u) \leq I (γ (t)) = φ (t)$ (respectively $\geq$ ).

3.1step 2.1F3

Fermat's interior extremum theorem [F3] applied to $φ$ at the interior point $0$ gives $φ^{'} (0) = 0$ , that is, $D I (u) h = 0$ .

4.1step 3.1A1∎

Since $h \in \ker D G (u)$ was arbitrary, $D I (u)$ vanishes on all of $\ker D G (u)$ , which is the assertion; the Axiom of Choice was used only through [F1] [A1].

Depends on

Used by

Dependency tree · two levels

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text) (standard reference, not scraped)
Riccardo Cristoferi, Calculus of Variations: Lecture Notes, Carnegie Mellon University 2016 (complete 133-page notes) (standard reference, not scraped)