CauchyegoSchwarz

CauchyegoSchwarz refers to the Cauchy–Schwarz inequality, a foundational result in linear algebra and analysis. The inequality states that for any vectors u and v in an inner product space, the absolute value of their inner product does not exceed the product of their norms: |⟨u,v⟩| ≤ ||u||·||v||. Equality occurs precisely when u and v are linearly dependent (one is a scalar multiple of the other). In Euclidean n-space with the standard dot product, this reads |∑ x_i y_i| ≤ sqrt(∑ x_i^2) sqrt(∑ y_i^2).

The inequality extends beyond finite-dimensional spaces. It has an integral form: for square-integrable functions f and g on a measure space, |∫ f g| ≤ sqrt(∫ f^2) sqrt(∫ g^2). In probability theory, with random variables X and Y of finite second moments, |E[XY]| ≤ sqrt(E[X^2] E[Y^2]). These forms underscore the role of CauchyegoSchwarz in bounding correlations and projections.

Consequences of the inequality are widespread. It implies the triangle inequality for norms, supports projections in inner product spaces, and underpins many estimates in analysis, geometry, statistics, and numerical methods. It also clarifies when equality holds, namely when the involved functions or vectors are proportional almost everywhere or linearly dependent.

Historically, the inequality is attributed to Augustin-Louis Cauchy and Hermann Schwarz and is sometimes named the Cauchy–Bunyakovsky–Schwarz inequality in various regions. It remains a central tool across mathematics, providing a unifying bound for inner products, integrals, and expectations.