A theorem which states that the absolute value of the dot product between two vectors is less than or equal to the product of the magnitudes of the two vectors
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(f^2)(g^2). 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, ((ai, bi))^2 (ai)^2(bi)^2. In addition to functional analysis, these inequalities have important applications in statistics and probability theory