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

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 inequality ⓘ linked to: Cauchy–Schwarz inequality Cauchy–Bunyakovsky inequality ⓘ linked to: Cauchy–Schwarz inequality Cauchy–Bunyakovsky–Schwarz inequality ⓘ linked to: Cauchy–Schwarz inequality Schwarz inequality ⓘ linked to: Cauchy–Schwarz inequality
appliesTo	Euclidean spaces ⓘ linked to: Euclidean space 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 inequality for sums ⓘ linked to: Cauchy–Schwarz inequality
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 theorem ⓘ linked to: Parseval's 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

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 Cauchy → notableFor → Cauchy–Schwarz inequality ⓘ

subject linked to: Augustin-Louis

Augustin-Louis Cauchy → notableFor → Cauchy’s inequality ⓘ

subject linked to: Augustin-Louis