Cauchy-Schwarz inequality (thing) by ariels

Also known as the Bunyakovski inequality, particularly in Russian-speaking countries.

This has many forms, but they all involve the comparison of an inner product or a hermitian product with the norms of the components. These forms are particularly useful:

For all reals a₁, ... , a_n, b₁, ... , b_n,
(a₁*b₁ + ... + a_n*b_n)² ≤ (a₁² + ... + a_n²) * (b₁² + ... + b_n²)
In an inner product space or a Hilbert space, (x,y) ≤ ||x||*||y||.
For all random variables X,Y, with standard deviations σ(X), σ(Y), their covariance is bounded by
Cov(X,Y) ≤ σ(X)*σ(Y)

A proof of the Cauchy-Schwarz inequality is elementary, if somewhat tricky.

Hölder's inequality generalizes the Cauchy-Schwarz inequality. A corollary of this inequality is that ||x||=sqrt((x,x)) satisfies the triangle inequality (for norms); the following proof is for real numbers only, but essentially the same proof works for Hilbert space, too.

||x+y||² = (x+y,x+y) = (x,x)+2(x,y)+(y,y) ≤ (x,x) + 2 sqrt((x,x) * (y,y)) + (y,y) = (||x||+||y||)²

A similar trick with Hölder's inequality gives Minkowski's inequality, for p-norms.

Proof of the Cauchy-Schwarz inequality	Hilbert space	Hölder's inequality	Minkowski's inequality
Jensen inequality	inner product	Heisenberg Uncertainty Principle	Cholesky factorisation
metric tensor	Formalism	Segmentation fault	Augustin-Louis Cauchy
hermitian conjugate	Cauchy's Inequality	covariance	Augustin Cauchy
AM-GM inequality	triangle inequality	Schwarz lemma	distance
inner product space	norm	Russian