CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Goldstine finite-data approximation

Statement

Assume HB. Let $X$ be a real or complex normed space, $x^{* *} \in B_{X^{* *}}$ , $f_{1}, \dots, f_{m} \in X^{*}$ , and $ε > 0$ . There exists $x \in B_{X}$ such that

$∣ f_{j} (x) - x^{* *} (f_{j}) ∣ < ε (1 \leq j \leq m) .$

The finite list may be empty.

Facts & Assumptions

Given: HB and the space, bidual vector, finite test list, and positive tolerance in the statement.

[F1]

Under HB, $J_{X} (B_{X})$ is weak-star dense in $B_{X^{* *}}$ (Goldstine's theorem).

[F2]

Finite evaluation inequalities with positive tolerance form basic weak-star neighborhoods, including the empty list (Basic weak star neighborhoods).

[F3]

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

Proof

technique · constructive

1.1F2construct

Define $U = {y^{* *} \in X^{* *} : ∣ y^{* *} (f_{j}) - x^{* *} (f_{j}) ∣ < ε for 1 \leq j \leq m}$ . It is a basic weak-star neighborhood of $x^{* *}$ ; when $m = 0$ , it is all of $X^{* *}$ .

2.1F1F3step 1.1

By Goldstine, $U$ meets $J_{X} (B_{X})$ , so there is $x \in B_{X}$ with $J_{X} (x) \in U$ . This invocation carries the HB hypothesis; no sequence or family of approximants is selected.

3.1step 2.1discharge-construct: step 1.1step 2.1∎

Since $J_{X} (x) (f_{j}) = f_{j} (x)$ , the witness $x$ from step 2.1 satisfies every displayed inequality, and hence is the required finite-data approximant.

Depends on

Used by

Milman–Pettis theorem Theorem

Dependency tree · two levels

9 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

Bühler–Salamon, Functional Analysis (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)