The Cauchy-Schwarz Inequality states that for any two vectors in an inner product space, the absolute value of their inner product is at most the product of their individual norms, with equality exactly when the vectors are linearly dependent. First proved by Augustin-Louis Cauchy for finite sums and later extended by Hermann Schwarz to integrals, it is one of the most widely used inequalities in mathematics, underlying results from probability theory to functional analysis.
Facts
StatementFor any two vectors in an inner product space, the absolute value of their inner product is at most the product of their individual norms, with equality exactly when the vectors are linearly dependent. 1 Classification
Statement Form Connections
Sources
1. Wikipedia: Cauchy-Schwarz inequality
Wikimedia FoundationLead section
The inequality for sums was published by Augustin-Louis Cauchy (1821).
Lead section, first sentence
The Cauchy-Schwarz inequality (also called Cauchy-Bunyakovsky-Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms.
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The Cauchy-Schwarz inequality (also called Cauchy-Bunyakovsky-Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms.
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