Cauchy–Schwarz inequality

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In mathematics, the Cauchy–Schwarz inequality (also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy–Bunyakovsky–Schwarz inequality), is a useful inequality encountered in many different settings, such as linear algebra, analysis, in probability theory, and other areas. It is a specific case of Hölder's inequality.

According to Michael Steele, the inequality is “one of the most widely used and most important inequalities in all of mathematics.”[1]

The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first stated by Viktor Bunyakovsky (1859) and rediscovered by Hermann Amandus Schwarz (1888) (often misspelled "Schwartz").

Contents

Statement of the inequality

The Cauchy–Schwarz inequality states that for all vectors x and y of an inner product space,

where \langle\cdot,\cdot\rangle is the inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as

Moreover, the two sides are equal if and only if x and y are linearly dependent (or, in a geometrical sense, they are parallel or one of the vectors is equal to zero).

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