Cauchy–Schwarz inequality

E239290

mathematical inequality result in analysis result in linear algebra

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)

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

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.

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 ⓘ