1.0 CAUCHY’S INEQUALITY

May 19th, 2015
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Cauchy–Schwarz inequality (also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy–Bunyakovsky–Schwarz inequality) is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas. It is considered to be one of the most important inequalities in all of mathematics.

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1.0 CAUCHYS INEQUALITYCauchySchwarz inequality (also known as the Bunyakovsky inequality, the Schwarz inequality, or the CauchyBunyakovskySchwarz inequality) is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas. It is considered to be one of the most important inequalities in all of mathematics. It has a number of generalizations, among them Hlder's inequality1.1STATEMENT OF THE INEQUALITYThe CauchySchwarz inequality states that for all vectors x and y of an inner product space,where is the inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written asMoreover, the two sides are equal if and only if x and y are linearly dependent (or, in a geometrical sense, they are parallel or one of the vectors is equal to zero).If and are any complex numbers and the inner product is the standard inner product then the inequality may be restated in a more explicit way as follows:When viewed in this way the numbers x1,,xn, and y1,,yn are the components of x and y with respect to an orthonormal basis of V.Even more compactly written:Equality holds if and only if x and y are linearly dependent, that is, one is a scalar multiple of the other (which includes the case when one or both are zero).1.2 PROOFLet u,v be arbitrary vectors in a vector space V over F with an inner product, whe

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