In mathematics , the Cauchy—Schwarz inequality , also known as the Cauchy—Bunyakovsky—Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra , analysis , probability theory , vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics. Examples of inner products include the real and complex dot product ; see the examples in inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as  .
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Let and be any two real integrable functions in , then Schwarz's inequality is given by. Schwarz's inequality is sometimes also called the Cauchy-Schwarz inequality Gradshteyn and Ryzhik , p. To derive the inequality, let be a complex function and a complex constant such that for some and.
Since , where is the complex conjugate ,. Writing this in compact notation,. Multiply 4 by and then plug in 5 and 6 to obtain. Abramowitz, M. New York: Dover, p. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp.
Buniakowsky, V. Gradshteyn, I. Tables of Integrals, Series, and Products, 6th ed. Hardy, G. Cambridge, England: Cambridge University Press, pp. Schwarz, H. Reprinted in Gesammelte Mathematische Abhandlungen, Vol. New York: Chelsea, pp. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
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