In mathematics, the Cauchy–Schwarz inequality (also known as the Bunyakovsky inequality, Schwarz inequality, Cauchy–Bunyakovsky–Schwarz inequality, or Cauchy–Bunyakovsky inequality), is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas. It is considered to be one of the most important inequalities in all of mathematics. It has a number of generalizations, among them Hölder's inequality.
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).
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“However energetically society in general may strive to make all the citizens equal and alike, the personal pride of each individual will always make him try to escape from the common level, and he will form some inequality somewhere to his own profit.”
—Alexis de Tocqueville (18051859)