TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Well-posedness of the adjoint variational problem

Statement

Assume Countable Choice. Let $V$ be a real or complex Hilbert space and let $a : V \times V \to K$ be linear in the first slot, conjugate-linear in the second, bounded by $M$ , and coercive with constant $α > 0$ : $∣ a (u, v) ∣ \leq M ∥ u ∥_{V} ∥ v ∥_{V} .$ $Re a (u, u) \geq α ∥ u ∥_{V}^{2} .$ For every bounded linear functional $G : V \to K$ , there is a unique $z \in V$ such that $a (v, z) = G (v) (v \in V),$ and $α ∥ z ∥_{V} \leq ∥ G ∥_{V^{*}}$ . In the real case conjugation is the identity and both forms and functionals are linear over $R$ .

Facts & Assumptions

Given: Countable Choice, a Hilbert space $V$ , the stated bounds for $a$ , and a bounded linear functional $G$ .

[F1]

A bounded coercive sesquilinear form is linear in the first slot, conjugate-linear in the second, and satisfies the displayed $M, α$ bounds (Bounded, coercive and symmetric sesquilinear forms).

[F2]

On a Hilbert space, a bounded coercive sesquilinear form with a bounded conjugate-linear right-hand side has a unique solution and the estimate $α ∥ z ∥ \leq ∥ F ∥$ (The Lax--Milgram theorem).

[F3]

Countable Choice is the axiom required by the imported Lax–Milgram theorem (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F1givenalgebra

Define the adjoint form by $a^{†} (z, v) : = \overline{a (v, z)}$ . The sesquilinearity of $a$ shows that $a^{†}$ is linear in $z$ and conjugate-linear in $v$ . Also $∣ a^{†} (z, v) ∣ = ∣ a (v, z) ∣ \leq M ∥ z ∥_{V} ∥ v ∥_{V}$ , and $Re a^{†} (z, z) = Re a (z, z) \geq α ∥ z ∥_{V}^{2}$ .

1.2givenalgebra

Set $G^{†} (v) : = \overline{G (v)}$ . Since $G$ is linear, $G^{†}$ is conjugate-linear, and $∣ G^{†} (v) ∣ = ∣ G (v) ∣ \leq ∥ G ∥_{V^{*}} ∥ v ∥_{V}$ ; hence $∥ G^{†} ∥_{V^{*}} = ∥ G ∥_{V^{*}}$ .

2.1F2F3step 1.1step 1.2∎

Apply [F2] to $a^{†}$ and $G^{†}$ . It gives a unique $z \in V$ with $a^{†} (z, v) = G^{†} (v)$ and $α ∥ z ∥_{V} \leq ∥ G^{†} ∥_{V^{*}}$ . Conjugating the equation gives $a (v, z) = G (v)$ for every $v$ , and the norm equality in step 1.2 gives the claimed estimate and uniqueness.

Depends on

Used by

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Sources

Thomas Richter, Die Finite Elemente Methode für partielle Differentialgleichungen (standard reference, not scraped)
Thomas Wick, Numerical Methods for Partial Differential Equations (standard reference, not scraped)