Triple

T17341158
Position Surface form Disambiguated ID Type / Status
Subject Banach–Saks theorem E421068 entity
Predicate relatedConcept P37 FINISHED
Object Banach–Saks property
The Banach–Saks property is a feature of certain Banach spaces where every bounded sequence has a subsequence whose Cesàro means converge in norm.
E421068 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: Banach–Saks property | Statement: [Banach–Saks theorem, relatedConcept, Banach–Saks property]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Banach–Saks property
Context triple: [Banach–Saks theorem, relatedConcept, Banach–Saks property]
  • A. Banach–Saks theorem
    The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
  • B. Eberlein–Šmulian theorem
    The Eberlein–Šmulian theorem is a fundamental result in functional analysis characterizing weak compactness in Banach spaces by showing that a subset is weakly compact if and only if it is weakly sequentially compact.
  • C. Banach limit
    A Banach limit is a linear functional on the space of bounded sequences that extends the usual limit and assigns generalized “limits” to sequences that may not converge in the classical sense.
  • D. Schreier family in Banach space theory
    The Schreier family in Banach space theory is a combinatorial collection of finite subsets of natural numbers introduced by Józef Schreier that plays a central role in constructing and analyzing special Banach spaces with unusual structural properties.
  • E. Banach–Stone theorem
    The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
  • 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: Banach–Saks property
Triple: [Banach–Saks theorem, relatedConcept, Banach–Saks property]
Generated description
The Banach–Saks property is a feature of certain Banach spaces where every bounded sequence has a subsequence whose Cesàro means converge in norm.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Banach–Saks property
Target entity description: The Banach–Saks property is a feature of certain Banach spaces where every bounded sequence has a subsequence whose Cesàro means converge in norm.
  • A. Banach–Saks theorem chosen
    The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
  • B. Eberlein–Šmulian theorem
    The Eberlein–Šmulian theorem is a fundamental result in functional analysis characterizing weak compactness in Banach spaces by showing that a subset is weakly compact if and only if it is weakly sequentially compact.
  • C. Banach limit
    A Banach limit is a linear functional on the space of bounded sequences that extends the usual limit and assigns generalized “limits” to sequences that may not converge in the classical sense.
  • D. Schreier family in Banach space theory
    The Schreier family in Banach space theory is a combinatorial collection of finite subsets of natural numbers introduced by Józef Schreier that plays a central role in constructing and analyzing special Banach spaces with unusual structural properties.
  • E. Banach–Stone theorem
    The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
  • 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_69d889d3adc881909319f1edb8d2a956 completed April 10, 2026, 5:25 a.m.
NER Named-entity recognition batch_69e43a15f6488190ad7d489e7391ab12 completed April 19, 2026, 2:12 a.m.
NED1 Entity disambiguation (via context triple) batch_6a018c588a7081909ab108cb4adfedfe completed May 11, 2026, 7:59 a.m.
NEDg Description generation batch_6a018e0f09c881909296656b2732bf1e completed May 11, 2026, 8:06 a.m.
NED2 Entity disambiguation (via description) batch_6a018e7b453c81909f75593237bcf9ec completed May 11, 2026, 8:08 a.m.
Created at: April 10, 2026, 5:44 a.m.