Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that the squared inner product of two vectors is at most the product of their squared lengths, with equality exactly when they are linearly dependent.

Inner product: An operation that assigns a scalar to two vectors and induces notions of length and angle. The inequality bounds the inner product relative to the lengths it defines.
Vector space: A set equipped with vector addition and scalar multiplication satisfying the vector-space axioms. The inequality applies to vectors in spaces with a suitable inner product.
Triangle inequality: The distance between two points is no greater than the sum of their distances to a third point. The inner-product bound is a standard step in proving the norm triangle inequality.
Minkowski inequality: The triangle inequality for Lp norms and finite-dimensional p-norms. It bounds the norm of a sum, whereas Cauchy–Schwarz bounds an inner product.
Augustin-Louis Cauchy: A French mathematician whose work shaped analysis, algebra, and mathematical physics. Cauchy established an early form of the inequality for finite sums.
Norm (mathematics): A function that assigns lengths to vectors and satisfies positivity, homogeneity, and the triangle inequality. An inner-product norm turns the bound into |⟨x,y⟩| ≤ ‖x‖‖y‖.
Euclidean space: A real vector space with the standard dot product and its associated distance and angles. The familiar dot-product version is the inequality's basic geometric setting.
Bunyakovsky inequality: The integral form of Cauchy–Schwarz bounds the squared integral of a product by the product of squared-integral norms. It extends the finite-dimensional bound to functions and integrals.
Young's inequality for products: A bound on a product by weighted powers, commonly ab ≤ aᵖ/p + bᵠ/q for conjugate exponents. It gives a scalar product bound that can replace Cauchy–Schwarz in estimates.
Viktor Bunyakovsky: A Russian mathematician who developed an integral version of the inequality in the nineteenth century. His work supplied the form that is also called the Cauchy–Bunyakovsky inequality.

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