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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, specifically in complex analysis, Cauchy's estimate gives local bounds for the derivatives of a holomorphic function. These bounds are optimal. These bounds are optimal. Cauchy's estimate is also called Cauchy's inequality , but must not be confused with the Cauchy–Schwarz inequality .
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.
In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a ...
In cases where the ideal linear system assumptions are insufficient, the Cauchy–Schwarz inequality guarantees a value of . If C xy is less than one but greater than zero it is an indication that either: noise is entering the measurements, that the assumed function relating x(t) and y(t) is not linear, or that y(t) is producing output due to ...
A general form, also known as the Cauchy–Binet formula, states the following: Suppose A is an m×n matrix and B is an n×m matrix. If S is a subset of {1, ..., n} with m elements, we write A S for the m×m matrix whose columns are those columns of A that have indices from S.
The Paley–Zygmund inequality is sometimes used instead of the Cauchy–Schwarz inequality and may occasionally give more refined results. Under the (incorrect) assumption that the events v , u in K are always independent, one has Pr ( v , u ∈ K ) = Pr ( v ∈ K ) Pr ( u ∈ K ) {\displaystyle \Pr(v,u\in K)=\Pr(v\in K)\,\Pr(u\in K)} , and ...
A vector field f : R n → R n is called coercive if ‖ ‖ + ‖ ‖ +, where "" denotes the usual dot product and ‖ ‖ denotes the usual Euclidean norm of the vector x.. A coercive vector field is in particular norm-coercive since ‖ ‖ (()) / ‖ ‖ for {}, by Cauchy–Schwarz inequality.