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The Lagrange multiplier rule for finitely many regular constraints

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a real Banach space (Banach space), let $U \subseteq X$ be open, let $I : U \to R$ be Fréchet differentiable at $u \in U$ , and let $G = (G_{1}, \dots, G_{m}) : U \to R^{m}$ be of class $C^{1}$ with $D G (u) : X \to R^{m}$ surjective (Fréchet derivative between Banach spaces). If $u$ is a local minimiser or a local maximiser of $I$ on the level set ${G = G (u)}$ , then there is a unique $λ \in R^{m}$ with $D I (u) = \sum_{i = 1}^{m} λ_{i} D G_{i} (u) .$

Facts & Assumptions

Given: A real Banach space $X$ , open $U \subseteq X$ , a functional $I$ Fréchet differentiable at $u$ , a $C^{1}$ map $G = (G_{1}, \dots, G_{m})$ with $D G (u)$ surjective, and the assumption that $u$ is a local minimiser or local maximiser of $I$ on the level set ${G = G (u)}$ .

[F1]

The differential annihilates the tangent kernel at a constrained extremum: under these hypotheses $D I (u) h = 0$ for every $h \in \ker D G (u)$ , that is, $⋂_{i = 1}^{m} \ker D G_{i} (u) \subseteq \ker D I (u)$ .

[F2]

Fréchet derivative between Banach spaces, The dual space X^* of a normed space and its dual norm: $D I (u)$ and each component $D G_{i} (u) = π_{i - 1} \circ D G (u)$ is a bounded linear functional on $X$ , and the kernel of $D G (u)$ is the intersection of the kernels of its components.

[F3]

Every natural-number-indexed list of nonempty sets has a choice function on its family of values: a finite family of nonempty sets indexed by a natural number admits a choice function.

[F4]

Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent: the functionals $ψ_{1}, \dots, ψ_{m} \in X^{*}$ are linearly independent exactly when $\sum c_{i} ψ_{i} = 0$ in $X^{*}$ forces all $c_{i} = 0$ .

[F5]

Functionals vanishing on a common kernel are combinations of an independent family: for $m \geq 1$ , if $ψ_{1}, \dots, ψ_{m} \in X^{*}$ are linearly independent and $⋂_{i = 1}^{m} \ker ψ_{i} \subseteq \ker φ$ for some $φ \in X^{*}$ , then there is a unique $λ \in R^{m}$ with $φ = \sum_{i = 1}^{m} λ_{i} ψ_{i}$ .

[F6]

The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ : $R^{0}$ is the zero space. For $m \geq 1$ , use the one-based labels $e_{i} : = {\hat{e}}_{i - 1}$ for $1 \leq i \leq m$ , where $({\hat{e}}_{k})_{k < m}$ is the supplied standard basis; likewise $G_{i} = π_{i - 1} \circ G$ and $λ_{i} = λ (i - 1)$ . Sums and intersections over $1 \leq i \leq m$ reindex those over $k < m$ ; at $m = 0$ the sum is the zero functional and the intersection of component kernels is $X$ .

[A1]

The Axiom of Choice: recorded as in the statement; the selections below are finite and need no choice principle.

Proof

technique · direct

Given: The hypotheses above, including the local extremum at $u$ .

1.1givenF1F2

By [F1] the differential $D I (u)$ vanishes on $\ker D G (u) = ⋂_{i = 1}^{m} \ker D G_{i} (u)$ [F2].

1.2givenF3F4F6choose

For $m \geq 1$ , the functionals $ψ_{i} : = D G_{i} (u) \in X^{*}$ are linearly independent. Indeed, since $D G (u)$ is surjective, for each $1 \leq i \leq m$ the preimage $D G (u)^{- 1} ({e_{i}})$ of the $i$ -th standard unit vector of [F6] is nonempty, so finite choice [F3], applied to the family indexed by $k < m$ with $i = k + 1$ , selects $x_{1}, \dots, x_{m} \in X$ with $D G (u) x_{i} = e_{i}$ , that is, $ψ_{j} (x_{i}) = δ_{i j}$ ; if $\sum_{i} c_{i} ψ_{i} = 0$ is the zero functional, evaluating at $x_{j}$ gives $c_{j} = 0$ for every $j$ , and independence follows by [F4].

2.1step 1.1step 1.2F5F6

If $m = 0$ , then $\ker D G (u) = X$ by [F6], and step 1.1 gives $D I (u) = 0$ . The unique vector of $R^{0}$ gives the zero empty sum, proving both existence and uniqueness of the multiplier identity. If $m \geq 1$ , apply [F5] with $ψ_{i} = D G_{i} (u)$ and $φ = D I (u)$ : the independence of step 1.2 and the kernel inclusion of step 1.1 are exactly its hypotheses, so there is a unique $λ \in R^{m}$ with $D I (u) = \sum_{i = 1}^{m} λ_{i} D G_{i} (u)$ .

3.1step 2.1A1∎

This is the asserted multiplier identity, with the uniqueness statement included; the constrained-stationarity supplier uses the implicit function theorem under the Axiom of Choice [A1], while the common-kernel argument above uses no additional choice principle.

Depends on

Used by

Dependency tree · two levels

45 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

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)