Triple

T17020364
Position Surface form Disambiguated ID Type / Status
Subject Schauder basis E412930 entity
Predicate relatedTo P37 FINISHED
Object Markushevich basis
A Markushevich basis is a biorthogonal system in a Banach space whose linear span is dense and whose associated functionals separate points, generalizing the notion of a Schauder basis.
E412930 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Markushevich basis | Statement: [Schauder basis, relatedTo, Markushevich basis]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Markushevich basis
Context triple: [Schauder basis, relatedTo, Markushevich basis]
  • A. Schauder basis
    A Schauder basis is a sequence in a Banach space such that every element of the space can be uniquely represented as a convergent infinite linear combination of the sequence’s vectors.
  • B. Riesz basis
    A Riesz basis is a sequence in a Hilbert space that is complete and behaves like an orthonormal basis up to a bounded, invertible linear transformation, allowing stable and unique expansions of vectors.
  • C. Gelfand–Tsetlin basis
    The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
  • D. Blaschke products
    Blaschke products are bounded analytic functions on the unit disk formed as (finite or infinite) products of Möbius transformations that map the disk to itself, playing a central role in complex analysis and function theory.
  • E. Bernstein polynomials
    Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Markushevich basis
Triple: [Schauder basis, relatedTo, Markushevich basis]
Generated description
A Markushevich basis is a biorthogonal system in a Banach space whose linear span is dense and whose associated functionals separate points, generalizing the notion of a Schauder basis.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Markushevich basis
Target entity description: A Markushevich basis is a biorthogonal system in a Banach space whose linear span is dense and whose associated functionals separate points, generalizing the notion of a Schauder basis.
  • A. Schauder basis chosen
    A Schauder basis is a sequence in a Banach space such that every element of the space can be uniquely represented as a convergent infinite linear combination of the sequence’s vectors.
  • B. Riesz basis
    A Riesz basis is a sequence in a Hilbert space that is complete and behaves like an orthonormal basis up to a bounded, invertible linear transformation, allowing stable and unique expansions of vectors.
  • C. Gelfand–Tsetlin basis
    The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
  • D. Blaschke products
    Blaschke products are bounded analytic functions on the unit disk formed as (finite or infinite) products of Möbius transformations that map the disk to itself, playing a central role in complex analysis and function theory.
  • E. Bernstein polynomials
    Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
  • F. None of above.

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69d886cc4170819093deddc7b8b4b6a7 completed April 10, 2026, 5:12 a.m.
NER Named-entity recognition batch_69e3d482c3a0819099e6ea4acb0a08ee completed April 18, 2026, 6:59 p.m.
NED1 Entity disambiguation (via context triple) batch_6a011b4f9dfc819085639edb5cda1cca completed May 10, 2026, 11:57 p.m.
NEDg Description generation batch_6a011cc1afc48190b83e3203407c1d7f completed May 11, 2026, 12:03 a.m.
NED2 Entity disambiguation (via description) batch_6a011d67c82c8190b737406e8952eb2b completed May 11, 2026, 12:05 a.m.
Created at: April 10, 2026, 5:33 a.m.