Bessel inequality
E825431
Bessel inequality is a fundamental result in functional analysis that bounds the sum of squared Fourier coefficients of a vector in an inner product space by the square of its norm.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Bessel inequality canonical | 2 |
| Bessel's inequality | 1 |
| Bessel’s inequality | 1 |
| Hilbert space inequality | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9843871 — 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: Bessel inequality Context triple: [Cauchy–Schwarz inequality, relatedTo, Bessel inequality]
-
A.
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
-
B.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
C.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
-
D.
Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
-
E.
Jensen inequality
Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
- 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: Bessel inequality Target entity description: Bessel inequality is a fundamental result in functional analysis that bounds the sum of squared Fourier coefficients of a vector in an inner product space by the square of its norm.
-
A.
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
-
B.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
C.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
-
D.
Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
-
E.
Jensen inequality
Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
inequality in inner product spaces
ⓘ
mathematical theorem ⓘ result in functional analysis ⓘ |
| appliesTo |
Hilbert space
ⓘ
linked to:
Hilbert spaces
inner product space ⓘ |
| assumes | orthonormality of the system (e_n) ⓘ |
| category |
Hilbert space inequality
ⓘ
linked to:
Bessel inequality
inequality involving inner products ⓘ |
| conclusion | series of squared Fourier coefficients is convergent and bounded by ||x||^2 ⓘ |
| doesNotRequire | completeness of the orthonormal system ⓘ |
| domainRestriction | orthonormal sequence (e_n) in an inner product space ⓘ |
| equalityCondition | orthonormal system is complete (Parseval identity holds) ⓘ |
| field |
Fourier analysis
ⓘ
Hilbert space theory ⓘ functional analysis ⓘ |
| generalizationOf | Pythagorean theorem for infinite orthogonal expansions ⓘ |
| holdsIn |
complex inner product spaces
ⓘ
real inner product spaces ⓘ |
| implies |
Fourier coefficients of x are square-summable
ⓘ
map x ↦ (⟨x,e_n⟩) is bounded from the space into ℓ² ⓘ |
| involvesConcept |
Fourier coefficients
ⓘ
Parseval identity ⓘ inner product ⓘ norm ⓘ orthonormal sequence ⓘ orthonormal system ⓘ series expansion ⓘ squared norm ⓘ |
| logicalForm | for all x and all orthonormal sequences (e_n), sum |⟨x,e_n⟩|^2 ≤ ||x||^2 ⓘ |
| mathematicalArea |
analysis
ⓘ
operator theory ⓘ |
| namedAfter | Friedrich Bessel ⓘ |
| relatedTo |
Cauchy–Schwarz inequality
ⓘ
Parseval theorem ⓘ
linked to:
Parseval's theorem
Riesz–Fischer theorem ⓘ |
| statementForm | sum |⟨x,e_n⟩|^2 ≤ ||x||^2 ⓘ |
| typeOfBound | upper bound on energy of Fourier coefficients ⓘ |
| usedFor |
bounding truncation error in Fourier series
ⓘ
establishing completeness criteria for orthonormal systems ⓘ proving Parseval identity ⓘ stability estimates in Hilbert space expansions ⓘ |
| usedIn |
Fourier series theory
ⓘ
linked to:
Fourier series
Fourier transform theory ⓘ
linked to:
Fourier transform
approximation theory ⓘ signal processing (theoretical foundations) ⓘ spectral theory of operators ⓘ |
| variable | vector x in an inner product space ⓘ |
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: Bessel inequality Description of subject: Bessel inequality is a fundamental result in functional analysis that bounds the sum of squared Fourier coefficients of a vector in an inner product space by the square of its norm.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bessel inequality
linked to: Bessel inequality
linked to: Bessel inequality