ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

A one-dimensional form attains the $1 / α$ Lax--Milgram bound

Example

On $H = C$ with the standard inner product and $α > 0$ , let $a (u, v) : = α u \overline{v}$ and let $F_{c} (v) : = c \overline{v}$ for a fixed $c \in C$ . Then $a$ is bounded with $M = α$ , coercive with the same constant $α$ , and the Lax--Milgram solution of $a (u, v) = F_{c} (v)$ for all $v$ is $u = \frac{c}{α},$ since $α u \overline{v} = c \overline{v}$ for all $v$ forces $α u = c$ . The solution operator has norm exactly $1 / α$ : $∥ F_{c} ∥ = \sup_{∣ v ∣ \leq 1} ∣ c \overline{v} ∣ = ∣ c ∣$ and $∣ u ∣ = ∣ c ∣ / α$ , so $∥ S ∥ = \frac{1}{α} .$ Hence the bound of The Lax--Milgram solution operator has norm at most $1 / α$ is attained and cannot be improved uniformly over coercive forms; this is the plan’s sharpness example and the one-dimensional model of the general estimate.

Facts & Assumptions

Given: A real $α > 0$ ; the Hilbert space $H = C$ with its usual inner product and modulus; the form $a (u, v) = α u \overline{v}$ ; and the functional $F_{c} (v) = c \overline{v}$ for a fixed $c \in C$ .

[F1]

$a$ is sesquilinear, bounded with $M = α$ , and coercive with the same constant: $∣ a (u, v) ∣ = α ∣ u ∣ ∣ v ∣$ and $a (u, u) = α ∣ u ∣^{2}$ (Bounded, coercive and symmetric sesquilinear forms, Real and imaginary parts, complex conjugation, and modulus, Hilbert space).

[F2]

Every conjugate-linear functional on $C$ has the form $F_{c} (v) = \overline{v} F_{c} (1)$ ; testing at $v = 1$ directly determines the unique solution. The abstract comparison is The Lax--Milgram solution operator has norm at most $1 / α$ , but no choice principle is needed for this scalar computation.

[F3]

Operator norm: $∥ F_{c} ∥ = \sup_{∣ v ∣ \leq 1} ∣ c \overline{v} ∣ = ∣ c ∣$ and $∥ S ∥ = \sup {∥ S (F) ∥ : ∥ F ∥ \leq 1}$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

Proof

1.1F1F2algebra

Direct solution: the equation $a (u, v) = F_{c} (v)$ reads $α u \overline{v} = c \overline{v}$ for every $v \in C$ . Testing with $v = 1$ forces $α u = c$ , that is $u = c / α$ ; conversely this $u$ satisfies the equation for every $v$ . The scalar equation also proves uniqueness directly.

1.2F1

Boundedness and coercivity constants: from $∣ a (u, v) ∣ = α ∣ u ∣ ∣ v ∣$ the least bound is $M = α$ , and coercivity holds with $α$ since $a (u, u) = α ∣ u ∣^{2}$ ; no larger coercivity constant can work at $u = 1$ .

2.1F2F3step 1.1

Norms: $∥ F_{c} ∥ = ∣ c ∣$ by [F3], and $S (F_{c}) = c / α$ has modulus $∣ c ∣ / α$ , so $∥ S (F_{c}) ∥ / ∥ F_{c} ∥ = 1 / α$ for every $c \neq 0$ ; hence $∥ S ∥ = 1 / α$ , attaining the bound of the corollary.

3.1step 1.2step 2.1∎

Conclusion: the estimate $∥ S ∥ \leq 1 / α$ is sharp and cannot be improved uniformly over bounded coercive forms on a fixed Hilbert space; the one-dimensional computation is the model of the general constant.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer Universitext, 2011, complete 614-page text) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford, complete 223-page author scan) (standard reference, not scraped)

A one-dimensional form attains the 1/α1/α Lax--Milgram bound