Alphabeta MathHow 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 or . For a normed with continuous dual from The dual space X^* of a normed space and its dual norm, the weak-star topology is the initial topology of all evaluations into with its usual topology. At , a neighbourhood basis consists of where is finite and . For the set is all of . 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
- A Reiter net has an invariant-mean cluster point Lemma
- Basic weak star neighborhoods Lemma
- C star state GNS construction, purity and Polish pure-state spaces Lemma
- Countable compactness closes in the bidual Lemma
- Dual ball as a closed subset of a product Lemma
- Irreducible representations separate arbitrary C star algebras Lemma
- Irreducible weak containment in a family selects one coefficient Lemma
- Probability-density approximation of continuous tests and topological means Lemma
- The fixed point property implies amenability Lemma
- BMO classes define bounded functionals on H1 Theorem
- Double annihilators give norm and weak-star closures Theorem
- Dunford--Pettis for real L¹ on a finite measure space Theorem
- Eberlein–Šmulian theorem Theorem
- Kernel-range identities and the weak-star closure of the transpose range Theorem
- Reflexive iff unit ball weakly compact Theorem
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.