DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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 weak-star topology from finite evaluations

Definition

Let $K = R$ or $C$ . For a normed $X$ with continuous dual $X^{*}$ from The dual space X^* of a normed space and its dual norm, the weak-star topology $σ (X^{*}, X)$ is the initial topology of all evaluations $f \mapsto f (x)$ into $K$ with its usual topology. At $f_{0}$ , a neighbourhood basis consists of $U (f_{0}; x_{1}, \dots, x_{n}; ε) = {f \in X^{*} : ∣ (f - f_{0}) (x_{j}) ∣ < ε (1 \leq j \leq n)},$ where $n$ is finite and $ε > 0$ . For $n = 0$ the set is all of $X^{*}$ . Finite intersections of inverse images of scalar open sets form the initial-topology basis; at the given point, finitely many disks can be refined using their smallest positive radius. Weak-star closure means closure in this topology, not merely sequential closure.

Depends on

Used by

Dependency tree · two levels

13 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, Example 3.9(i), p.122; Lemma 3.6(i), pp.119–120 (real source; complex disk version via the same initial-topology construction) (standard reference, not scraped)