Riesz basis
E747349
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Riesz basis canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8640759 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Riesz basis Context triple: [Frigyes Riesz, knownFor, Riesz basis]
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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.
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B.
Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
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C.
Bochner–Riesz means
Bochner–Riesz means are a family of summability methods in harmonic analysis used to improve the convergence of Fourier series and Fourier integrals by smoothing their partial sums.
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D.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
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E.
Riesz representation theorem
The Riesz representation theorem is a fundamental result in functional analysis that characterizes continuous linear functionals on Hilbert spaces as inner products with a unique vector in the space.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Riesz basis Target entity description: 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.
-
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–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
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C.
Bochner–Riesz means
Bochner–Riesz means are a family of summability methods in harmonic analysis used to improve the convergence of Fourier series and Fourier integrals by smoothing their partial sums.
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D.
Riesz
Riesz is a Hungarian surname most notably associated with the influential mathematician Frigyes Riesz, a pioneer in functional analysis.
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E.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
basis in functional analysis
ⓘ
mathematical concept ⓘ |
| appearsIn | monographs on frames and bases in Hilbert spaces ⓘ |
| characterizedBy |
equivalent to an orthonormal basis
ⓘ
image of an orthonormal basis under a bounded invertible operator ⓘ |
| definedOn | Hilbert space ⓘ |
| field |
Hilbert space theory
ⓘ
functional analysis ⓘ |
| generalizationOf | orthonormal basis up to bounded invertible operator ⓘ |
| hasComponent | Riesz basis constants A and B ⓘ |
| hasCondition |
bounded invertible analysis operator
ⓘ
bounded invertible synthesis operator ⓘ satisfies Riesz basis inequalities ⓘ sequence is complete in the Hilbert space ⓘ sequence is minimal ⓘ |
| hasCriterion | Bari theorem for Riesz bases close to orthonormal bases ⓘ |
| hasDual | unique biorthogonal sequence forming a Riesz basis ⓘ |
| hasInequality | A‖c‖² ≤ ‖∑ c_n x_n‖² ≤ B‖c‖² for some 0 < A ≤ B < ∞ ⓘ |
| hasProperty |
Gram matrix defines a bounded invertible operator on ℓ²
ⓘ
allows stable expansions ⓘ allows unique expansions ⓘ biorthogonal dual sequence exists and is unique ⓘ boundedly complete ⓘ complete sequence ⓘ coordinate functionals are continuous ⓘ equivalent norms via coefficient sequences ⓘ expansion coefficients depend continuously on the vector ⓘ not necessarily normalized ⓘ not necessarily orthogonal ⓘ perturbation stable under small bounded perturbations ⓘ topological basis in Hilbert space ⓘ unconditional if and only if equivalent to an orthonormal basis with unconditional convergence ⓘ |
| hasRelation | Riesz basis = complete Riesz sequence ⓘ |
| implies |
Schauder basis property
ⓘ
closed linear span equals whole Hilbert space ⓘ stability of coefficient functionals ⓘ unique coefficient representation for each vector ⓘ |
| namedAfter | Frigyes Riesz ⓘ |
| relatedTo |
Riesz sequence
ⓘ
Schauder basis ⓘ biorthogonal system ⓘ frame (functional analysis) ⓘ orthonormal basis ⓘ |
| usedIn |
numerical analysis of operator equations
ⓘ
partial differential equations ⓘ signal processing ⓘ spectral theory ⓘ time-frequency analysis ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Riesz basis Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.