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The 36 pages that link to Cauchy–Schwarz inequality, each with the reason it gives.

• Triangle inequalityRelated: Together with inner-product identities, it proves the triangle inequality for Euclidean distance.

• Inner productRelated: It makes the normalized inner product a valid cosine and underpins norm estimates.

• Dot productRelated: It constrains dot products and guarantees the cosine formula stays between −1 and 1.

• Euclidean normRelated: It constrains dot products using Euclidean lengths and yields the triangle inequality.

• Inner product spaceRelated: It ensures the inner-product norm obeys the triangle inequality.

• Gram matrixRelated: It bounds each Gram entry by the corresponding diagonal entries.

• Riesz representation theoremRelated: It proves that inner product with a fixed vector defines a bounded functional.

• Equality conditionRelated: Its equality condition is linear dependence of the vectors, illustrating a structural criterion.

• Geometric inequalityRelated: Vector representations of geometric quantities often reduce a proof to this bound.

• Harmonic meanRelated: It yields the relationship between the harmonic and arithmetic means for positive values.

• Rearrangement inequalityRelated: It supplies a complementary bound on product sums without relying on sorted pairings.

• Vector geometryRelated: It bounds angles and lengths and underlies geometric inequalities for vectors.

• AM–GM inequalityRelated: Repeated applications yield common proofs of AM–GM for powers of two terms.

• Bessel's inequalityRelated: It supplies foundational estimates used throughout inner-product-space analysis.

• Lax–Milgram theoremRelated: It is a basic estimate behind boundedness of inner-product-defined forms.

• Ptolemy's inequalityRelated: It is a broader inequality often used in analytic proofs, while Ptolemy's is specifically a four-point distance relation.

• Euclidean vectorRelated: It constrains possible angles and underpins the Euclidean distance formula.

• Pedoe's inequalityRelated: Vector inequalities of this type provide a route to bounding the area terms.

• Kantorovich inequalityRelated: Inner-product inequalities provide a route to bounding the paired quadratic forms.

• Shapiro inequalityRelated: Cauchy–Schwarz gives a direct lower bound for the cyclic sum, though not always the sharp one.

• Szemerédi's theoremRelated: Repeated applications of this inequality are central to many proofs and counting arguments.

• QM–AM–GM–HM inequalitiesRelated: Applied to the numbers and to their reciprocals, it establishes the quadratic-mean and harmonic-mean bounds.