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Consider an inner product space in the sense of a vector space equipped with a Hermitian form which is positive semi-definite:
(we do not need to assume positive definiteness, cf. MO:a/2548691, Ćurgus)
then for all pairs of vectors the following Cauchy-Schwarz inequality holds:
In terms of the norm and the absolute value this means equivalently:
Original proofs are due to Cauchy in 1821, Bouniakowsky in 1859, Hermann Schwarz in 1888.
Review:
Branko Ćurgus, Cauchy-Bunyakovsky-Schwarz inequality
Wikipedia, Cauchy-Schwarz inequality
Last revised on January 14, 2024 at 09:12:13. See the history of this page for a list of all contributions to it.