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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:
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 implies the inner product is jointly continuous in norm and can therefore be extended to the completion. The action of A {\displaystyle A} on E {\displaystyle E} is continuous: for all x {\displaystyle x} in E {\displaystyle E}
One particularly useful inequality to analyze homomorphism densities is the Cauchy–Schwarz inequality. The effect of applying the Cauchy-Schwarz inequality is "folding" the graph over a line of symmetry to relate it to a smaller graph. This allows for the reduction of densities of large but symmetric graphs to that of smaller graphs.
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. [8]
By the Cauchy–Schwarz inequality, ... [12] [13] An important class of matrix-valued reproducing kernels are separable kernels which can factorized as the product of ...
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. Cambridge University Press. ISBN 978-0-521-54677-5 .