ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

The Riesz representative of an explicit functional on $C^{3}$

Example

For the linear functional

$f (z_{0}, z_{1}, z_{2}) = (1 + i) z_{0} - 2 i z_{1} + 3 z_{2}$

on standard $C^{3}$ , the Riesz representative under the linear-first convention is

$w = (1 - i, 2 i, 3),$

so $f (z) = ⟨ z, w ⟩$ .

Facts & Assumptions

Given: The displayed linear functional $f$ .

[L1]

Every functional on a finite-dimensional inner product space has a unique vector $w$ such that $f (z) = ⟨ z, w ⟩$ (Finite-dimensional Riesz representation: every functional is uniquely $v \mapsto ⟨ v, w ⟩$ ).

[L2]

The standard complex inner product is $⟨ z, w ⟩ = \sum_{k} z_{k} \overline{w_{k}}$ (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

Verification

technique · computation

1.1L1L2algebra

By [L2], $⟨ z, (1 - i, 2 i, 3) ⟩ = (1 + i) z_{0} - 2 i z_{1} + 3 z_{2} = f (z)$ . Uniqueness in [L1] identifies the displayed vector as the representative.

2.1step 1.1L1L2∎

If the functional is multiplied by $i$ , its representative is $- i w$ , since $⟨ z, - i w ⟩ = i ⟨ z, w ⟩$ . This explicitly exhibits the conjugate-linear dependence asserted by [L1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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.

The Riesz representative of an explicit functional on C3C3