Cauchy-Schwarz inequality

Cauchy-Schwarz inequality
/koh"shee shwawrts", koh shee"shvahrts"/, Math.

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Any of several related inequalities developed by Augustin-Louis Cauchy and, later, Herman Schwarz (1843–1921).

The inequalities arise from assigning a real number measurement, or norm, to the functions, vectors, or integrals within a particular space in order to analyze their relationship. For functions f and g, whose squares are integrable and thus usable as a norm, (∫fg)2/n≤/n(∫f2)(∫g2). For vectors a = (a1, a2, a3,..., an) and b = (b1, b2, b3,..., bn), together with the inner product (see inner product space) for a norm, (Σ(ai, bi))2 ≤ Σ(ai)2Σ(bi)2. In addition to functional analysis, these inequalities have important applications in statistics and probability theory.

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      Any of several related inequalities developed by Augustin-Louis Cauchy (Cauchy, Augustin-Louis, Baron) and, later, Herman Schwarz (1843–1921). The inequalities arise from assigning a real number measurement, or norm, to the functions, vectors, or integrals within a particular space in order to analyze their relationship. For functions f and g, whose squares are integrable and thus usable as a norm, (∫fg)2 ≤ (∫f2)(∫g2). For vectors a = (a1, a2, a3,…, an) and b = (b1, b2, b3,…, bn), together with the inner product (see inner product space) for a norm, (Σ(ai, bi))2 ≤ Σ(ai)2Σ(bi)2. In addition to functional analysis, these inequalities have important applications in statistics and probability theory.

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Universalium. 2010.

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