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Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps [27]) — For a 2-positive map between C*-algebras, for all , in its domain, () ‖ ‖ (), ‖ ‖ ‖ ‖ ‖ ‖. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:
He is credited with an early discovery of the Cauchy–Schwarz inequality, proving it for the infinite dimensional case in 1859, many years prior to Hermann Schwarz's research on the subject. Bunyakovsky is an author of Foundations of the mathematical theory of probability (1846). [7] Bunyakovsky published around 150 research papers. [1]
In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, [1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu ...
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.
The Cauchy–Schwarz inequality asserts that ... This inequality is sharp, with equality achieved when x or X is a Dirac mass, ...
Cauchy–Schwarz inequality: An upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.
1 Cauchy–Schwarz inequality. 2 On a complex Hilbert space, if an operator is non-negative then it is symmetric. 3 If an operator is non-negative and defined on the ...
Cauchy's inequality may refer to: the Cauchy–Schwarz inequality in a real or complex inner product space Cauchy's estimate , also called Cauchy's inequality, for the Taylor series coefficients of a complex analytic function