Cauchy–Schwarz inequality
E239290
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.
All labels observed (7)
| Label | Occurrences |
|---|---|
| Cauchy–Schwarz inequality canonical | 7 |
| Cauchy inequality | 1 |
| Cauchy inequality for sums | 1 |
| Cauchy–Bunyakovsky inequality | 1 |
| Cauchy–Bunyakovsky–Schwarz inequality | 1 |
| Cauchy’s inequality | 1 |
| Schwarz inequality | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171651 — 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: Cauchy–Schwarz inequality Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy–Schwarz inequality]
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
-
E.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
- 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: Cauchy–Schwarz inequality Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
-
E.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in analysis ⓘ result in linear algebra ⓘ |
| alsoKnownAs |
Cauchy–Schwarz inequality
ⓘ
surface form:
Cauchy inequality
Cauchy–Schwarz inequality ⓘ
surface form:
Cauchy–Bunyakovsky inequality
Cauchy–Schwarz inequality ⓘ
surface form:
Cauchy–Bunyakovsky–Schwarz inequality
Cauchy–Schwarz inequality ⓘ
surface form:
Schwarz inequality
|
| appliesTo |
Euclidean space
ⓘ
surface form:
Euclidean spaces
Hilbert spaces ⓘ inner product spaces ⓘ |
| category |
inequalities in probability theory
ⓘ
inequalities in vector spaces ⓘ |
| coreStatement | For any vectors u and v in an inner product space, |⟨u,v⟩| ≤ ∥u∥ ∥v∥ ⓘ |
| equalityCondition |
equality holds if and only if the two vectors are linearly dependent
ⓘ
equality holds if one vector is a scalar multiple of the other ⓘ |
| expressedIn |
complex inner product spaces
ⓘ
real inner product spaces ⓘ |
| field |
functional analysis
ⓘ
geometry ⓘ linear algebra ⓘ probability theory ⓘ real analysis ⓘ |
| generalizationOf |
Cauchy inequality for integrals
ⓘ
Cauchy–Schwarz inequality self-linksurface differs ⓘ
surface form:
Cauchy inequality for sums
|
| hasFormulation |
For functions f and g in L^2, |∫ f g| ≤ (∫ |f|^2)^{1/2} (∫ |g|^2)^{1/2}
ⓘ
For sequences (a_i) and (b_i), (∑ a_i b_i)^2 ≤ (∑ a_i^2)(∑ b_i^2) ⓘ |
| implies |
non-negativity of inner product norms
ⓘ
triangle inequality in inner product spaces ⓘ |
| namedAfter |
Augustin-Louis Cauchy
ⓘ
Hermann Amandus Schwarz ⓘ |
| relatedTo |
Bessel inequality
ⓘ
Hölder inequality ⓘ Jensen inequality ⓘ Minkowski inequality ⓘ Parseval's theorem ⓘ
surface form:
Parseval theorem
|
| underpins |
concept of correlation in statistics
ⓘ
definition of angle between vectors in inner product spaces ⓘ geometry of Hilbert spaces ⓘ |
| usedFor |
bounding covariances in probability theory
ⓘ
bounding integrals in functional analysis ⓘ deriving correlation coefficient bounds ⓘ error estimation in numerical analysis ⓘ establishing orthogonality properties ⓘ proving Bessel inequality ⓘ proving Hölder inequality ⓘ proving Minkowski inequality ⓘ proving Parseval identity ⓘ proving convergence of series and integrals ⓘ proving the triangle inequality for norms induced by inner products ⓘ |
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: Cauchy–Schwarz inequality Description of subject: 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.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy
subject surface form:
Augustin-Louis Cauchy
this entity surface form:
Cauchy’s inequality
this entity surface form:
Cauchy inequality
this entity surface form:
Cauchy–Bunyakovsky–Schwarz inequality
this entity surface form:
Cauchy–Bunyakovsky inequality
this entity surface form:
Schwarz inequality
this entity surface form:
Cauchy inequality for sums