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.

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All labels observed (7)

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

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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
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- If the subject is not a named entity, return an empty list.
- Include at least one triple where predicate is "instanceOf".
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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.

Augustin-Louis Cauchy knownFor Cauchy–Schwarz inequality
Hölder inequality generalizes Cauchy–Schwarz inequality
Hölder inequality implies Cauchy–Schwarz inequality
Inequalities containsTopic Cauchy–Schwarz inequality
Augustin-Louis notableFor Cauchy–Schwarz inequality
subject surface form: Augustin-Louis Cauchy
Augustin-Louis notableFor Cauchy–Schwarz inequality
subject surface form: Augustin-Louis Cauchy
this entity surface form: Cauchy’s inequality
Cauchy–Schwarz inequality alsoKnownAs Cauchy–Schwarz inequality
this entity surface form: Cauchy inequality
Cauchy–Schwarz inequality alsoKnownAs Cauchy–Schwarz inequality
this entity surface form: Cauchy–Bunyakovsky–Schwarz inequality
Cauchy–Schwarz inequality alsoKnownAs Cauchy–Schwarz inequality
this entity surface form: Cauchy–Bunyakovsky inequality
Cauchy–Schwarz inequality alsoKnownAs Cauchy–Schwarz inequality
this entity surface form: Schwarz inequality
Cauchy–Schwarz inequality generalizationOf Cauchy–Schwarz inequality self-linksurface differs
this entity surface form: Cauchy inequality for sums
Hadamard inequality relatedTo Cauchy–Schwarz inequality
Hadamard inequality proofTechnique Cauchy–Schwarz inequality