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Mathematical inequality relating inner products and norms. The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) [1][2][3][4] is 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 ...
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}
Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, [4] this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce ...
In optics, Cauchy's transmission equation is an empirical relationship between the refractive index and wavelength of light for a particular transparent material. It is named for the mathematician Augustin-Louis Cauchy , who originally defined it in 1830 in his article "The refraction and reflection of light".
Online book chapter Hilbert’s Inequality and Compensating Difficulties extracted from Steele, J. Michael (2004). "Chapter 10: Hilbert's Inequality and Compensating Difficulties". The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press. pp. 155–165. ISBN 0-521-54677-X..
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 ...
The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. [1] Hölder's inequality holds even if ‖ fg ‖ 1 is infinite, the right-hand side also being infinite in that case. Conversely, if f is in L p (μ) and g is in L q (μ), then the pointwise product fg is in L 1 (μ).