ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

Nested trial spaces give nonincreasing energies, sometimes with equality

Example

For $H = diag (- 1, 1, 3)$ on $C^{3}$ , take the nested trial spaces $K_{1} = span {e_{2}}$ , $K_{2} = span {e_{2}, e_{3}}$ , and $K_{j} = C^{3}$ for every $j \geq 3$ . Their minimum unit-vector energies $\min_{v \in K_{j}, ∥ v ∥ = 1} v^{*} H v$ are $1, 1, - 1, - 1, \dots$ . The sequence is nonincreasing but need not decrease strictly at every enrichment; its limit is the infimum over the union.

Facts & Assumptions

Given: $H = diag (- 1, 1, 3)$ and the nested subspaces $K_{1} \subset K_{2} \subset K_{3} = K_{4} = \dots$ displayed above.

[F1]

Nested compact trial sets for a common continuous energy have nonincreasing minima and converge to the infimum over their union (Nested finite-basis infima decrease to the infimum over their union).

[F2]

A Hermitian matrix has an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F3]

Closed bounded subsets of finite-dimensional real coordinate space are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F4]

A continuous real-valued function on a nonempty compact set attains its minimum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Verification

1.1F2F3F4given

Each unit sphere in $K_{j}$ is closed and bounded in finite-dimensional real coordinates, hence compact by [F3]. The energy $v \mapsto v^{*} H v$ is continuous, so [F4] gives a minimum on each sphere; diagonalizing the restrictions by [F2] gives the values $1, 1, - 1$ on $K_{1}, K_{2}, K_{3}$ , respectively, and $- 1$ for every $j \geq 3$ .

2.1F1step 1.1algebra∎

The spaces satisfy $K_{1} \subset K_{2} \subset K_{3}$ and their infinite union is $C^{3}$ . The computed values are nonincreasing, with equality at the first enrichment, and converge to $- 1 = \inf_{v \in C^{3}, ∥ v ∥ = 1} v^{*} H v$ . Applying [F1] to the infinite nested sequence of their unit spheres gives this same limit.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

50 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.