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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Galerkin orthogonality for conforming approximations

Statement

Let $V$ be a vector space over $K \in {R, C}$ , let $V_{h} \subseteq V$ be a subspace, let $a$ be linear in its first argument and conjugate-linear in its second, and let $F : V \to K$ be conjugate-linear. If $u \in V$ and $u_{h} \in V_{h}$ satisfy $a (u, v) = F (v) (v \in V), a (u_{h}, v_{h}) = F (v_{h}) (v_{h} \in V_{h}),$ then $a (u - u_{h}, v_{h}) = 0 for every v_{h} \in V_{h} .$

Facts & Assumptions

Given: The common form and functional, the inclusion $V_{h} \subseteq V$ , and the two displayed variational equations.

Proof

1.1givenchoose

Fix $v_{h} \in V_{h}$ . Since $V_{h} \subseteq V$ , the continuous equation is valid at $v_{h}$ , so $a (u, v_{h}) = F (v_{h})$ .

2.1givenstep 1.1algebra∎

The discrete equation gives $a (u_{h}, v_{h}) = F (v_{h})$ . Subtracting the two equalities and using the stated linearity of $a$ in its first slot yields $a (u - u_{h}, v_{h}) = 0$ .

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VI, Chapter 26, Basic error analysis (standard reference, not scraped)
Thomas Richter, Die Finite Elemente Methode für partielle Differentialgleichungen (standard reference, not scraped)