CauchySchwarz

Cauchy-Schwarz, formally the Cauchy–Schwarz inequality, is a fundamental result in mathematics that provides an upper bound for the absolute value of the inner product of two vectors by the product of their norms. In any inner product space over the real or complex numbers, for all vectors u and v one has |⟨u,v⟩| ≤ ||u|| ||v||, equivalently ⟨u,v⟩^2 ≤ ⟨u,u⟩⟨v,v⟩. In complex spaces the left-hand side uses the magnitude of the inner product.

Equality occurs precisely when the two vectors are linearly dependent, i.e., one is a scalar multiple of the other. This makes the inequality tight and characterizes exact alignment of the vectors.

The inequality has broad formulations and applications. In finite-dimensional vector spaces it bounds dot products; in analysis it appears as the integral form |∫ f g| ≤ √(∫ f^2) √(∫ g^2) for square-integrable functions; in probability it underpins bounds on covariance and correlation. It is a key ingredient in proofs of the triangle inequality, Hölder’s inequality, and in establishing that Gram matrices are positive semidefinite.

Beyond its standard form, there are numerous generalizations to different fields, to function spaces, and to operator versions for matrices and linear functionals. The Cauchy–Schwarz inequality is a staple result across linear algebra, analysis, probability, and related areas, reflecting the geometric intuition that the projection of one vector onto another cannot exceed the product of their lengths.