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.