Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-10-22 (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.

Existence of an x0-minimum-norm exact solution

Statement

Assume the ultrafilter lemma, dependent choice, and HB. Let X,Y be real Hilbert spaces, let D⊆X be nonempty and weakly closed in the coarsest topology making every bounded real-linear functional continuous, and let F:D→Y satisfy: for every sequence xn∈D and x∈X,z∈Y, if xn⇀x and F(xn)⇀z, then x∈D and F(x)=z; weak convergence means convergence under every bounded real-linear functional. If SF(y):={x∈D:F(x)=y} is nonempty for some y∈Y, then for every x0∈X there exists x†∈SF(y) such that ∥x†−x0∥X≤∥x−x0∥X for every x∈SF(y).

Facts & Assumptions

Given: The ultrafilter lemma, DC, and HB; real Hilbert spaces X,Y; a weakly closed forward map F:D→Y; a nonempty exact solution set SF(y); and x0∈X.

[A1]

The ultrafilter lemma says every filter on a set extends to an ultrafilter (The ultrafilter extension principle (UL/BPI)).

[A2]

DC supplies an infinite sequence following any everywhere-extendible relation (The axiom of dependent choice: a relation in which every element is related to something admits an N-indexed chain).

[A3]

HB is the real dominated-extension principle (The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

The exact solution set SF(y) is weakly sequentially closed (The exact solution set of a weakly closed map is weakly sequentially closed).

[F2]

DC implies countable choice, the principle that every countable family of nonempty sets admits a choice function (Dependent choice implies countable choice, The Axiom of Countable Choice (ACω)); every complete real inner-product space is reflexive under that principle (Hilbert spaces are reflexive by Riesz representation).

[F3]

On a nonempty weakly sequentially closed subset of a real reflexive Banach space, a proper coercive weakly sequentially lower-semicontinuous functional attains its infimum under the ultrafilter lemma, DC, and HB (The direct method in a reflexive Banach space).

[F4]

Under HB, weak convergence implies lower semicontinuity of the norm (Weak convergence implies lower semicontinuity of the norm).

[F5]

An extended-real functional on a nonempty admissible set is proper when it is finite somewhere; coercivity is equivalent to bounded sublevel sets; weak sequential lower semicontinuity is the liminf inequality along weakly convergent sequences (Proper, coercive and weakly lower semicontinuous extended-real functionals).

[F6]

An x0-minimum-norm solution is an exact solution whose distance to x0 is no greater than that of any other exact solution in D (x0-minimum-norm solutions on weakly closed domains).

Proof

Given: The assumptions above.

1.1givenF1F5construct

Let S:=SF(y). It is nonempty by hypothesis and weakly sequentially closed by [F1]. Define I:S→(−∞,+∞] by I(x):=12∥x−x0∥X2. This is finite everywhere on S, so it is proper. Its sublevel set {x∈S:I(x)≤Λ} is empty for Λ<0 and, for Λ≥0, satisfies ∥x∥X≤∥x0∥X+2Λ by the triangle inequality. Hence I is coercive on S.

2.1A3F4F5step 1.1algebra

If xn∈S and xn⇀x∈S, then xn−x0⇀x−x0. By [F4], ∥x−x0∥X≤lim inf⁡n∥xn−x0∥X; since norms are nonnegative and squaring is increasing and continuous there, I(x)≤lim inf⁡nI(xn). Thus I is weakly sequentially lower semicontinuous on S.

3.1A1A2A3F2F3step 1.1step 2.1construct

By [F2], countable choice follows from the assumed DC, so the real Hilbert space X is reflexive; it is Banach by completeness in the Hilbert-space definition. The nonempty set S is weakly sequentially closed by step 1.1, and I is proper, coercive, and weakly sequentially lower semicontinuous by steps 1.1 and 2.1. Apply [F3], whose choice assumptions are exactly [A1], [A2], and [A3], to obtain x†∈S minimizing I on S.

4.1F6step 3.1algebra∎

Since I(x)=12∥x−x0∥X2 and the square function is increasing on nonnegative reals, minimizing I on S is equivalent to minimizing ∥x−x0∥X. Thus x† satisfies [F6] and is an x0-minimum-norm exact solution.

Remarks

The theorem gives existence, not uniqueness: a nonlinear exact solution set need not have a unique point at minimum distance from x0. The use of DC also supplies the countable choice assumed by the Hilbert reflexivity supplier; no full axiom of choice is used.

Depends on

Used by

Dependency tree · two levels

57 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